Subextensions for a permutation $\mathrm{PSL}_2(q)$-module

Using cohomological methods, we solve the problem of embedding $\mathrm{SL}_2(q)$ into the permutation wreath product for the permutation $\mathrm{PSL}_2(q)$-module in characteristic $2$ that arises from the action on the projective line. We also prove some useful auxiliary results.