Let $m:G\times X\to X$ be a $G$-group, i.e the an action which is a group homomorphism. From this answer I learned that the semidirect product $G\rtimes_mX$ is the homotopy quotient $X/\!/_{\!m}G$.
I was wondering - does the knit product $G\underset{\alpha,\beta}\bowtie H$ also admit an elegant homotopy colimit characterization? Does it describe (isomorphism classes of) short exact sequences which are split on both sides?
