Equivariant Hochschild cohomology of group algebras and relative $\operatorname{Ext}$

Abstract: For a finite group $Γ$, acting on a finite group $G,$ we find necessary conditions for which the first $Γ_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $Γ_0$ is the stabilizer subgroup in $Γ$ of some element in $G.$ For any field $k$ we show that the $Γ$-equivariant Hochschild cohomology of $Γ$-algebras with coefficients in a $Γ$-equivariant bimodule (Jensen, 1996) is isomorphic with some $kΓ$-relative $\operatorname{Ext},$ in the context of relative homological algebra.