Suppose a group G acts on a chain complex K and induced action on H(K) is trivial. What "secondary operations" on H(K) can be defined in this situation?

**Example.** If $G=\langle\sigma\rangle/\sigma^n$ acts trivially on H(K) then $x-\sigma x=dl(x)$ (for some function $l$) and a secondary operation $x\mapsto l(x)+\sigma l(x)+\dots+\sigma^{n-1}l(x)$ is well-defined mod n. And this operation is non-trivial (consider a complex $Z[G]\to Z[G]$, $x\mapsto (1-\sigma)x$).

So looks like these operations has something to do with group homology, but details elude me.

**Update.** Two nice answers explain what is the meaning of the operation from the example above (and how it can be defined for an arbitrary group).

But does this construction give *all* operations? I.e. what structure on H(K) one needs to recover K (up to q/iso)? Like,

- associative multiplication on K $\Leftrightarrow$ $A_\infty$-structure on H(K);
- G-action on K $\Leftrightarrow$ ??? on H(K).

(Perhaps, there is a very general answer: not just for k[G] but for an arbitrary algebra — or even arbitrary operad, maybe. Probably, Tyler Lawson's comment is relevant — if somebody could elaborate on that...)

`$A_\infty$`

structures, the action of G on K is recoverable from either an`$A_\infty$`

-action of k[G] on K, or an`$A_\infty$`

-map from k[G] to End(K). $\endgroup$