Let $p>2$ be a prime integer and let $\mathbb{F}\_{p^2}$ be the corresponding finite field. Consider a subgroup $H$ of $SL\_2(\mathbb{F}\_{p^2})$ which satisfies the following conditions:


1. The matrix $\left(\begin{smallmatrix} 0& 1\\ -1 &0 \end{smallmatrix}\right)\in H$.
2. At least half of all elements in $H$ have trace $0$.


(Edited: I just realize that in fact I have even more conditions on $H$:


3. $H$ is not Abelian.
4. The trace of every element in $H$ is in fact in $\mathbb{F}\_p$.)


My question is: Can we classify the isomorphism types of the image of $H$ in $PGL\_2(\mathbb{F}\_{p^2})$?


I have done some MAGMA calculation for $p=3,5,7$. It seems that the quotient images of $H$ satisfying conditions (1)-(4) are either


1. A cyclic group.
2. Dihidral group.
3. Semiproduct of a cyclic group and a dihidral group. 
I am wondering if this is always true.


(PS: I asked the same question at [Math Stackexchange](https://math.stackexchange.com/questions/3449887/special-type-of-subgroups-of-sl-2-mathbbf-p2) since I am not sure if this question is trivial. According to comments, I will delete the one in math exchange.)



$PGL_2$ image of special subgroups in $SL_2(\mathbb{F}_{p^2})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let $H<\SL\_2(p^2)$ for $p$ an odd prime. Let $Z=Z(\SL\_2(p^2))$. The only important supposition is that half of all elements of $H$ have trace $0$. Let $h\in H$ be such an element.


As Mark Wildon [observes](https://mathoverflow.net/questions/346848/pgl-2-image-of-special-subgroups-in-sl-2-mathbbf-p2#comment868342_346848), the fact that $h$ has trace $0$, and is in $\SL\_2(p^2)$, means that it has characteristic polynomial $\lambda^2+1$. This means that $h$ is diagonalizable in $\SL\_2(p^2)$, and that $h^2=-I$. In particular $hZ$ is an involution in $\PSL\_2(p^2)$ -- note that $\PSL\_2(p^2)$ has a unique conjugacy class of these. Thus at least half of the elements in the group $HZ/Z<\PSL\_2(p^2)$ are involutions.


Now we make use of the classification of subgroups of $\PSL\_2(p^2)$ which is usually attributed to Dickson:


*Dickson, Leonard Eugene*, Linear groups. With an exposition of the Galois field theory. With an introduction by Wilhelm Magnus. Unabridged and unaltered republ. of the first ed, New York: Dover Publications, Inc. XVI, 312 p. (1958). [ZBL0082.24901](https://zbmath.org/?q=an:0082.24901).


(Note that Bray-Holt-Roney Dougal also cite E.H. Moore and Wiman for early work on this result. Lots more information can be found in Michael Guidici's [preprint on the subject, "Maximal subgroups of almost simple groups with socle $\PSL(2, q)$"](https://arxiv.org/abs/math/0703685).)


Anyway, one can go through the list given by Dickson and see directly that one of the following holds:


* $HZ/Z$ is cyclic of order $2$;
* $HZ/Z$ is dihedral of order $r$ where $r$ divides $2p$, $p^2-1$ or $p^2+1$;
* $HZ/Z$ is isomorphic to $E\rtimes C\_2$ where $E$ denotes an elementary abelian group of order $p^2$ ($E$ is a Sylow $p$-subgroup of $\PSL\_2(p^2)$), and the non-identity element in the $C\_2$ acts on $E$ by inversion.


Note that I consider the Klein 4-group to be a dihedral group of order $4$ in the list above.


**Final remark**: Using Dickson's result is probably overkill. As I mentioned [above](https://mathoverflow.net/questions/346848/pgl-2-image-of-special-subgroups-in-sl-2-mathbbf-p2#comment869388_346848), C. T. C. Wall has classified all groups where at least half of the elements are involutions ([Wall - On groups consisting mostly of involutions](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-groups-consisting-mostly-of-involutions/554A7AE2A5C7B537B76CB098A8C6E2E2) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=249503))). It seems likely that this result could be used instead. (And there are probably even more elementary methods.)


