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3 added 2 characters in body; added 9 characters in body

Let's assume it is a chain complex of vector spaces over some field. Write the action as $g\mapsto a(g)$ where $a:K\to K$ is a chain map. For each $g\in G$, $a(g)$ is chain homotopic to the identity map. Choose $b(g)$ such that $db(g)+b(g)d=a(g)-1$. Then $db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$ and $da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if $c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$ c(g,h)=b(g)a(h)-a(g)b(h)-b(g)+b(h)$then$dc(g,h)+c(g,h)d=0$. This gives an element of$Hom(H_nK,H_{n+1})K$Hom(H_nK,H_{n+1}K)$ for every $g$ and $h$, and I believe a well-defined element of $H^2(G;Hom(H_nK,H_{n+1}K))$, for every $n$.

Edit: This is the same sort of thing that Tyler got in his answer. I was thinking about it like this: Imagine that it makes sense to speak of the topological group of automorphisms of $K$. We have a map of $G$ into $ker(Aut(K)\to \pi_0Aut(HK))$ pi_0(Aut(K))=Aut(HK))$and thus a map from BG$BG$into the classifying space of the latter. This classifying space is simply connected and has$\pi_2=\pi_1Aut(K)=\pi_1End(K)=\prod_n Hom(H_nK,H_{n+1}K)$2 added 424 characters in body Let's assume it is a chain complex of vector spaces over some field. Write the action as$g\mapsto a(g)$where$a:K\to K$is a chain map. For each$g\in G$,$a(g)$is chain homotopic to the identity map. Choose$b(g)$such that$db(g)+b(g)d=a(g)-1$. Then$db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$and$da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if$c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$then$dc(g,h)+c(g,h)d=0$. This gives an element of$Hom(H_nK,H_{n+1})K$for every$g$and$h$, and I believe a well-defined element of$H^2(G;Hom(H_nK,H_{n+1}K)$, H^2(G;Hom(H_nK,H_{n+1}K))$, for every $n$.

Edit: This is the same sort of thing that Tyler got in his answer. I was thinking about it like this: Imagine that it makes sense to speak of the topological group of automorphisms of $K$. We have a map of $G$ into $ker(Aut(K)\to \pi_0Aut(HK))$ and thus a map from BG into the classifying space of the latter. This classifying space is simply connected and has $\pi_2=\pi_1Aut(K)=\pi_1End(K)=\prod_n Hom(H_nK,H_{n+1}K)$

1

Let's assume it is a chain complex of vector spaces over some field. Write the action as $g\mapsto a(g)$ where $a:K\to K$ is a chain map. For each $g\in G$, $a(g)$ is chain homotopic to the identity map. Choose $b(g)$ such that $db(g)+b(g)d=a(g)-1$. Then $db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$ and $da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if $c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$ then $dc(g,h)+c(g,h)d=0$. This gives an element of $Hom(H_nK,H_{n+1})K$ for every $g$ and $h$, and I believe a well-defined element of $H^2(G;Hom(H_nK,H_{n+1}K)$, for every $n$.