This didn't seem likely, so I looked for a counterexample and found one after a short search. Clearly $Q$ must not have any normal Sylow subgroups, so the first example to try is $Q=S_4$. Then $N$ must not be abelian, so I tried $N=Q_8$ and found an example.

The example is $\mathtt{SmallGroup}(192,988)$ in GAP or Magma. This has seven generators $x_1,\ldots,x_7$, and the normal subgroup isomorphic to $Q_8$ is $\langle x_2, x_3x_7 \rangle$.

Checking that the extension is nonsplit but it splits over the normalizers of Sylow $2$- and $3$-subgroups is routine in GAP or Magma, but I can tell you the commands if it would help.