I am trying to find out what results are already out there in this direction.
For example, from the Ito-Michler theorem, if $\rho(G)$ is the set of prime divisors of irreducible characters of $G$ and either $\rho(G)\subseteq \pi$ or $|\rho(G)\cap\pi|\leq 1$ then we are guaranteed the existence of a $\pi$-Hall subgroup.
Also, there is a result by J. Thompson that if some prime $p$ divides the degrees of all non-linear irreducible characters of a finite groups, then that group has a normal $p$-complement.


Are there other results like these, which give a relations between some set connected to the degrees of irreducible characters of a group and a set $\pi$ of prime numbers, such that we are guaranteed the existence of a $\pi$-Hall subgroup?



Existence of certain Hall-subgroups based on knowledge of the degrees of irreducible characters
Background:



> 
> (Ito–Michler) p does not divide the order of the degree of any (absolutely) irreducible (ordinary) character of G iff G has a normal, abelian, Sylow p-subgroup.
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For any subset π of ρ(G)′, it follows that G has a normal, abelian, Hall π-subgroup, and hence also a Hall π′-subgroup. Flipping the roles of π and π′ gives your first existence claim.


Chapter 14 of Isaacs's Character Theory textbook contains this and further results, which I assume are already known. For instance, the classical:



> 
> (Blichfeldt) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > n, G has an abelian, Hall π-subgroup.
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Blichfeldt's paper below contains the result that G has an abelian, normal, Hall π-subgroup for π a collection of primes p > (n-1)(2n+1). This was refined to:



> 
> (Feit–Thompson) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > 2n+1, G has an abelian, normal, Hall π-subgroup.
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> 
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The more detailed information alluded to in Isaacs's textbook is likely Winter's 1964 paper below. It gives a complete list of groups with non-normal Sylow p-subgroups for p > (2n+1)/3. A non-abelian but normal Sylow π-subgroup is implied by the following:



> 
> (Feit) if G has a a faithful, irreducible (ordinary) character of degree p-2 ≥3, then either the Sylow p-subgroup of G is normal, G is a direct product of SL(2,p) and an abelian group, or G is a direct product of 3.Alt(6) and abelian group.
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Now each of these results required *faithful* characters. If you are OK with that, then Berkovich ([MR1189113](http://www.ams.org/mathscinet-getitem?mr=1189113)) has a Thompson like hypothesis and an Ito like conclusion (normal Hall π-subgroup containing the nilpotent residual), but it looks at π(G/Z(χ)) too.


The Ito–Michler result is also valid in some sense over finite fields:



> 
> (Manz) Every irreducible, p-modular character degree is coprime to p iff G has a normal Sylow p-subgroup.
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> (Manz–Wolf) If G is p-solvable and every irreducible, p-modular character degree is coprime to q, then Oq′(G), the subgroup generated by the Sylow q-subgroups, is solvable of q-length at most 2 (π-length roughly measures how far away from having a normal Hall π-subgroup you are).
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(Navarro, et al.) has generalized both the Ito and Thompson results to blocks in [MR1810119](http://www.ams.org/mathscinet-getitem?mr=1810119), [MR1956546](http://www.ams.org/mathscinet-getitem?mr=1956546), and [MR2159761](http://www.ams.org/mathscinet-getitem?mr=2159761).


A more recent article by (Matterei) gives a reasonable survey of results. The way he rephrases results may be quite interesting to you. He also gives the Carter–Hawkes ([MR1199667](http://www.ams.org/mathscinet-getitem?mr=1199667)) results that continue the ideas of Thompson. A recent paper of Dolfi et. al ([MR2469367](http://www.ams.org/mathscinet-getitem?mr=2469367)) derives Ito-like results using the orders of a non-vanishing elements g in G, such that χ(g) ≠ 0 for any any ordinary, irreducible character χ and derives Ito's and Thompson's results as corollaries.




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Blichfeldt, H. F. "On the order of linear homogeneous groups."
Trans. AMS. 4 (1903), 387–397. [JFM 34.0176.02](http://www.zentralblatt-math.org/zmath/en/search/?q=an:34.0176.02) [JSTOR 1986408](http://www.jstor.org/stable/1986408)


Feit, Walter; Thompson, John G. "Groups which have a faithful representation of degree less than (p-1)/2". Pacific J. Math. 11 (1961), 1257–1262. [MR133373](http://www.ams.org/mathscinet-getitem?mr=133373) [euclid.pjm/1103036911](http://projecteuclid.org/getRecord?id=euclid.pjm/1103036911)


Winter, David L. "Finite groups having a faithful representation of degree less than (2p+1)/3." Amer. J. Math. 86 (1964), 608–618. [MR183788](http://www.ams.org/mathscinet-getitem?mr=183788) [DOI: 10.2307/2373026](http://dx.doi.org/10.2307/2373026)


Feit, Walter. "On finite linear groups. II." J. Algebra 30 (1974), 496–506. [MR357634](http://www.ams.org/mathscinet-getitem?mr=357634) [DOI: 10.1016/0021-8693(74)90220-8](http://dx.doi.org/10.1016/0021-8693(74)90220-8)


Manz, Olaf. "On the modular version of Ito's theorem on character degrees for groups of odd order." Nagoya Math. J. 105 (1987), 121–128. [MR881011](http://www.ams.org/mathscinet-getitem?mr=881011) [euclid.nmj/1118780642](http://projecteuclid.org/getRecord?id=euclid.nmj/1118780642)


Manz, Olaf; Wolf, Thomas R. "Brauer characters of q′-degree in p-solvable groups."
J. Algebra 115 (1988), no. 1, 75–91. [MR937602](http://www.ams.org/mathscinet-getitem?mr=937602) [DOI: 10.1016/0021-8693(88)90283-9](http://dx.doi.org/10.1016/0021-8693(88)90283-9)


Mattarei, Sandro. "Retrieving information about a group from its character degrees or from its class sizes." Proc. Amer. Math. Soc. 134 (2006), no. 8, 2189–2195. [MR2213690](http://www.ams.org/mathscinet-getitem?mr=2213690) [DOI: 10.1090/S0002-9939-06-08274-8](http://dx.doi.org/10.1090/S0002-9939-06-08274-8)


