# cohomology: is the following variant of the universal coefficient theorem valid?

A version of the Theorem of the universal coefficient that relates the entire cohomology of a group $$G$$ To his cohomology with coefficients in an abelian group. $$M$$ It is as follows:

$$H ^ n (G, M) = H ^ n (G, mathbb Z) otimes M times quad {text} _1 ^ { mathbb Z} (H ^ {n + 1} ( G, mathbb Z), M)$$

It is assumed in this expression that $$G$$ Acts trivially on the coefficients. Now suppose $$G$$ It is an extension of the group. $$mathbb Z_2$$ by some group $$G_0$$ Y $$M = mathbb Z_2$$. I am interested in finding the cohomology of $$G$$ With coefficients in the module. $$mathbb Z ^ {sgn}$$, whose coefficients are skewed by the $$mathbb Z_2$$ factor, while $$G_0$$ It has trivial action. I wanted to know if the following statement, which looks like the statement of the UCT, is really valid:

$$H ^ n (G, mathbb Z_2) = H ^ n (G, mathbb Z ^ {sgn}) otimes mathbb Z_2 quad text {Tor} _1 ^ { mathbb Z} ( H ^ {n + 1} (G, mathbb Z ^ {sgn}), mathbb Z 2)$$

(So, to be clear, l.h. has a trivial action in $$mathbb Z_2$$, while the r.h.s. has non-trivial action in $$mathbb Z ^ {sgn}$$. I've checked it by hand to see some simple examples and it works, but I have no proof. )