$PGL_2$ image of special subgroups in $SL_2(\mathbb{F}_{p^2})$ 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:


*

*The matrix $\left(\begin{smallmatrix} 0& 1\\ -1 &0 \end{smallmatrix}\right)\in H$. 

*At least half of all elements in $H$ have trace $0$. 


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


*$H$ is not Abelian. 

*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


*

*A cyclic group.

*Dihidral group.

*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 since I am not sure if this question is trivial. According to comments, I will delete the one in math exchange.)
 A: $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$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, 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.
(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 $\PSL(2, q)$".)
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, C. T. C.  Wall has classified all groups where at least half of the elements are involutions (Wall - On groups consisting mostly of involutions (MSN)). It seems likely that this result could be used instead. (And there are probably even more elementary methods.)
