Let $G$ be a finite group, $H \subset G$ is a subgroup and $K = G/H$ is the quotient group, so we have extension of groups $$ 1 \to H \to G \to K \to 1. $$ Let $R$ be a commutative ring, can one build group algebra $R[G]$ out of the group algebras $R[H]$ and $R[K]$?
In the trivial case $G\cong H \times K$ we have $R[G] \cong R[H] \otimes_R R[K]$, but how one can generalize this for non-trivial extensions?