Automorphisms of nonsplit coverings of $PS{L}_2(q)$ in odd characteristic dividing $q-1$

We classify the nonsplit extensions of elementary abelian $p$-groups by $PS{L}_2(q)$, with odd $p$ dividing $q-1$, for an irreducible induced action, calculate the relevant low-dimensional cohomology groups, and describe the automorphism groups of such extensions.