Let’s define the set of *outer-commutator group words* $OC \subset F\_\infty = F[x\_0, x\_1, …, x\_n, …]$ using the following recurrence:


$$\forall i \in \mathbb{N} \text{ } x\_i \in OC$$


$$\forall u, v \in OC \text{ } [u, v] \in OC$$


Let’s call a group variety *outer commutator variety* iff it can be defined by a single outer commutator identity. Examples of outer commutator varieties include the varieties of 


a) all $n$-step nilpotent groups (defined by $[…[[x\_0, x\_1], x\_2]… x\_n]$)


b) all $n$-Engel groups (defined by $[…[[x\_0, x\_1], x\_1]… x\_1]$)


c) all $n$-step soluble groups (defined by $[…[[x\_0, x\_1], [x\_2, x\_3]]…, ...[[x\_{n-1}, x{n-2}],[x\_{n-1}, x\_n]]…]$


d) the trivial group (defined by $x\_1$)


e) all groups (defined by $[x\_1, x\_1]$)


Also, the class of outer commutator varieties is closed under variety product.


There are three theorems proved by Eugeny Khukhro and Natalya Makarenko about outer commutator varieties:


**First Khukhro-Makarenko Theorem**



> 
> Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $H \in \mathfrak{U}$ and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $N \in \mathfrak{U}$.
> 
> 
> 


**Second Khukhro-Makarenko Theorem**



> 
> Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $|V\_\mathfrak{U}(H)| < \infty$ and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $|V\_\mathfrak{U}(H)| < \infty$.
> 
> 
> 


**Third Khukhro-Makarenko Theorem**



> 
> Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $V\_\mathfrak{U}(H)$ is locally finite and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $V\_\mathfrak{U}(H)$ is locally finite.
> 
> 
> 


Here $V\_\mathfrak{U}$ stands for the corresponding verbal subgroup.


For the First Khukhro-Makarenko theorem, there is an upper bound proved by Anton Klyachko and Yulia Melnikova:


If $\mathfrak{U}$ is defined by an outer commutator group word $w$, then


$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!)$$


where 


$$f(x) = x(x+1)$$


$$f^{(n)}(x) = \begin{cases} x & \quad n = 0 \\ f(f^{(n-1)}(x)) & \quad n > 0 \end{cases}$$


$$d(x\_i) = 1$$


$$d([v, u]) = d(v) + d(u)$$


For the conditions of the second Khukhro-Makarenko theorem a similar bound can be derived:


$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!) + ln(|V\_\mathfrak{U}(H)|)$$


My question is:


Are there any similar upper bounds for $[G: N]$ under conditions of the Third Khukhro-Makarenko theorem?


Personally, I failed to find anything for that third case.


Yesterday, I [asked this question on MSE](https://math.stackexchange.com/q/3399454/407165) and was advised to re-ask it here.



Bounds for Khukhro-Makarenko theorems
The Second, Third, and many others "Khukhro--Makarenko theorems" were proved
[here](http://dx.doi.org/10.1016/j.jalgebra.2014.10.029), see also
[arXiv](http://arxiv.org/abs/1309.0571) (so, they are rather 
"Klyachko--Milentyeva theorems"). 
The bound is always the same:
$$
\log\_2|G:N|\leqslant f^{d(w)-1}(\log\_2(|G:H|!)).
$$
(In your question, there are several misprints in the estimates.)
These facts are special cases of *The Large-Subgroup Theorem* from the paper cited above. 


