Some necessary conditions for compatibility of groups

Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G\mathrm{/}{N}_1$ and $L_2\cong G\mathrm{/}{N}_2$. In this paper, we give new necessary conditions for two groups to be compatible.