Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, there are exactly three groups $G$ with $N<G<P$.
Question: Is it possible that these three groups are all isomorphic?
Comments: There are plenty of examples satisfying the hypothesis. The smallest is $P=C_4\times C_2$, with $M=1\times C_2$ and $N=C_2\times 1$. In this case, of the three groups between $N$ and $P$, two are isomorphic to $C_4$ and one is isomorphic to $C_2^2$. Using magma, I have checked all groups $P$ of order at most $128$ and it seems that the answer to the question is "no" for these (barring a mistake in my code, which is entirely possible).