Consider groups $G_i, K_i, F_i$ such that $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ , for $i = 1, 2$. In each case, find an example with the given properties or prove that no such example exists. If you give an example, justify why $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ and the various (non)isomorphisms.
a) $G_1 \ncong G_2$ and $K_1 \cong K_2$ and $F_1\cong F_2.$
I know that if we set $G_1 = G_2$ and $K_1 = K_2$ and $F_1 = F_2$, then they're all false. But, I need to find a specific example where it's possible, or if it is impossible. I'm not sure which ones are impossible, because I can't seem to come up with any examples.
$\triangleleft$ is for normal subgroup, and $\cong$ is for isomorphic here.