Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha',\beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\,{}_{\alpha'}\!\! \bowtie_{\beta'} \,(G,*)$. Moreover, if we fix the group $H$ and the automorphism $\sigma \in \Aut(H)$ then any $\sigma$-invariant isomorphism $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for
bicrossed product of groups are given.