Cohomology of PSL₂(q)

In 2011, Guralnick and Tiep proved that if G was a Chevalley group with Borel subgroup B and V an irreducible G-module in cross characteristic with V^B=0, then the dimension of H¹(G,V) is determined by the structure of the permutation module on the cosets of B. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if H≤G such that O_r^′(H)=O^r(H) and V^H=0 for V a G-module in characteristic r then dim⁡H¹(G,V) is determined by the structure of the permutation module on cosets of H, and Hⁿ(G,V) by Ext_Gⁿ⁻¹(V^⁎,M) for some kG-module M dependent on H. We also determine Ext_Gⁿ(V,W) for all irreducible kG-modules V, W for G∈{PSL₂(q),PGL₂(q),SL₂(q)} in cross characteristic.