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. )