Frattini's Argument: If $G$ is a finite group with normal subgroup $H$, and if $P$ is a Sylow $p$-subgroup of $H$, then $G=N_G (P)H$, where $N_G (P)$ denotes the normalizer of $P$ in $G$.
Recall that a subgroup $H$ of $G$ is called a subnormal subgroup of $G$ if there exists a chain $H=H_0 \leq H_1\leq \cdots \leq H_r =G$ of subgroups of $G$ so that $H_i$ is normal in $H_{i+1}$ for all i.
Is the above statement true when $H$ is a subnormal subgroup of $G$?
No.
For the smallest counterexample take $G=D_8$ the dihedral group of order $8$ given by $G=\langle x,y | y^4=x^2=1, x^{-1}yx=y^{-1}\rangle$. Let $H=\langle x\rangle$ which is subnormal. We have $P=H$ and $N_G(P)=\langle x, y^2 \rangle$.