Assume that $G$ is a finite group. Let $k$ be a field. Let $\varepsilon$ be the augmentation $kG\rightarrow k$. Consider the following map
$\varepsilon\otimes id:k[G]\otimes_k k[G]\rightarrow k[G]$
since $k[G]\otimes_kk[G]$ is isomorphic to $k[G\times G]$ this allows us to view $k[G]$ as a $k[G\times G]$-module. My question is: is this module a projective $k[G\times G]$-module?
It is clear when $char\;k\not\mid|G|$ since in this case $k[G\times G]$ is semisimple so everything is projective, but what happens in general? Is that module still projective?