At quick glance my following solution seems fine, and will edit it otherwise:

Take the short exact sequence of modules $\mathbb{Z}\hookrightarrow Ind^\Gamma_G\mathbb{Z}\twoheadrightarrow\mathbb{Z}$ and apply $H_i(\Gamma,-)$.  Note that this sequence is exact because $|\Gamma:G|=2$.  You obtain the long exact sequence, in particular:

$H_2\Gamma\stackrel{\delta}{\rightarrow}H_1\Gamma\stackrel{tr}{\rightarrow}H_1G\stackrel{res}{\rightarrow}H_1\Gamma$.

Exactness implies $Ker(tr)=Im(\delta)=H_2\Gamma/Ker(\delta)$, so that $|Ker(tr)|\le |H_2\Gamma|$.  And we know that $tr=Ver$ and $H_2\Gamma=M(\Gamma)$.


- Chris Gerig