TITLE: "Secondary operations" for a group acting on a chain complex
QUESTION [12 upvotes]: 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...)

REPLY [3 votes]: (Nothing really new here, but just to link $A_\infty$-description and the explicit construction.)
Action of G on K induces $A_\infty$-action of Z[G] on H(K)=:H. And this is all operations one can define on H(K), since this structure defines K up to (equivariant) q/iso (ref).
Now, let's rewrite this in more concrete terms. $A_\infty$-action of Z[G] on H is just a collection of maps $m_i\colon G^{i-1}\times H\to H$ aka $m_i\colon G^{i-1}\to\operatorname{Hom}(H,H)$ ($i=2,3,\dots$). This maps are subject to some relations. In particular, if $m_{i+1}$ is first non-zero higher action, then $m_{i+1}$ satisfies cocycle condition and gives an element in $H^i(G,\operatorname{Hom}(H,H))$. For i=2 it is exactly the element from Tom Goodwillie's answer (and, probably, it also coincides with $d_i$ from Tyler Lawson's answer, although I don't have a proof).