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

For a finite group $\mathrm{\Gamma }$, acting on a finite group $G,$ we find necessary conditions for which the first ${\mathrm{\Gamma }}_0$-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 ${\mathrm{\Gamma }}_0$ is the stabilizer subgroup in $\mathrm{\Gamma }$ of some element in $G\mathrm{.}$ For any field $k$ we show that the $\mathrm{\Gamma }$-equivariant Hochschild cohomology of $\mathrm{\Gamma }$-algebras with coefficients in a $\mathrm{\Gamma }$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\mathrm{\Gamma }$-relative $\mathrm{Ext},$ in the context of relative homological algebra.