Let $G$ be a finite group. The Wielandt subgroup of $G$ is defined to be the intersection of all the normalizers of the subnormal subgroups of $G$ i.e. $$w(G) = \bigcap_{H \triangleleft \triangleleft G} N_G(H)$$ Now we can define the Lower Wielandt Series by the descending normal series given by $w_0(G) = G$, $w_1(G)= w(G)$, $$w_{\alpha +1}(G) = \hspace{-3mm}\bigcap \limits_{K\in \;\Omega_{\alpha}(G)}\hspace{-3mm}w(K)$$ where $\Omega_{\alpha}(G)$ is the set of subgroups of $G$ that contain $w_\alpha(G)$ and $$w_{\lambda}(G) = \bigcap \limits_{\beta < \lambda}w_\beta(K) \;\;\text{ if $\lambda$ is limit ordinal}$$ We denote the last term of this series by $\bar w(G)$
Question: Suppose that we have the subgroup $H\bar w(G)$ for some $H\unlhd G$. Is it true that $\bar w(G) \subset w(H\bar w(G))?$
Now, I know that $\bar w(G)$ is subnormal in $G$ and since $\bar w(G) \leq H\bar w(G)$, $\bar w(G)$ is subnormal in $H\bar w(G)$. I'm not sure how to proceed. Any possible guidance will be appreciated. I have asked this question MSE and nobody responded thus I have posted it on this forumd
