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Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups.

We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$.

What is the homotopy quotient of $k[B]$ by $k[A]$ computed in differential graded algebras?

Here by homotopy quotient I mean the homotopy coequalizer of the pair consisting of $k[\iota] : k[A] \to k[B]$ and the map $1 : k[A] \to k[B]$ sending $a$ to $1 \in B$.

Can this homotopy coequalizer be equivalent to $k[C]$?

If it helps: $A$ can be assumed to be free.

Any hints are appreciated.

Thank you.


Note: I have now gained a somewhat better understanding of this question which has lead me to rather ask the following question: Stability property for differential graded algebras

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