I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. Math. 6 (1954), 274–304.

This is a group of order $2^7 3^5 5 = 155520$. Its commutator subgroup $H=(G,G)$ is of index 3, and a computer calculation shows that $H$ is isomorphic to ${\rm Sp}(4,3):={\rm Sp}_4({\Bbb F}_3)$, the symplectic group of of $4\times 4$ matrices over the finite field ${\Bbb F}_3$.

This group $G$ is given with a faithful 4-dimensional complex representation $$\rho: G\to {\rm GL}(4, {\Bbb C}).$$ Moreover, it is is stable under the standard complex conjugation in ${\Bbb C}^4$, and so we obtain an involutive automorphism (an automorphism of order 2) $\ \sigma\colon H\to H$.

I am trying to *guess* this involution $\sigma$ and to compute the first nonabelian cohomology set $H^1(\langle\sigma\rangle, H)$. A computer calculation shows that the $H^1$ is trivial, and I would like to understand this without computer.

Question 1.What are the nontrivial 4-dimensional complex representations of the finite group ${\rm Sp}(4,3)$?

Question 2.What are the involutive automorphisms of ${\rm Sp}(4,3)$ ? In particular, is it true that all nontrivial involutive automorphisms of ${\rm Sp}(4,3)$ come from elements of order 2 in the projective symplectic group ${\rm PSp}_4({\Bbb F}_3)$ ?

Question 3.Which of those involutive automorphisms of $H={\rm Sp}(4,3)$ can come from the complex conjugation in a 4-dimensional complex representation of $H$?

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