$H_1\triangleleft G_1$, $H_2\triangleleft G_2$, $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2 \nRightarrow G_1\cong G_2$ Find a counterexample to show that if $ G_1 $ and $G_2$ groups,
$H_1\triangleleft G_1$, $H_2\triangleleft G_2$, $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2 \nRightarrow G_1\cong G_2$
I tried but I did not have success, I believe that these groups are infinite.

The standard counter example to that implication is the quaternion group $Q_8$ and dihedral group $D_4$.
Both groups have order $2^3=8$, so that every maximal group (i.e. one of order 4) is normal. The cyclic group of order 4 is contained in both groups and the quotient has order 2 in both cases. So all assertions are satisfied but $D_4\ncong Q_8$, of course.
