Let $G$ be a locally compact group with the group of topological group automorphisms $Aut(G)$ furnished with the compact-open topology. Let $B$ be a subgroup of $Aut(G)$. We call $G$ an [IN$]\_B$ if there is a $B$-invariant relatively compact neighbourhood of the identity element of the group $G$. 


It is known that (e.g. look at [1, Theorem 2.5]) for an [IN$]\_B$ group $G$, the intersection of all $B$-invariant relatively compact neighbourhoods of the identity forms a compact subgroup of $G$, denoted here by $K\_B$.


*I am looking for a group $G$ with a subgroup $B$ in $Aut(G)$ so that $K\_B$ is not a normal subgroup.* 


Remarks:


1. One can easily note that the group $B$ cannot include all the inner automorphisms.
2. By [1, Theorem 0.1], we can conclude that $B$ cannot be relatively compact in $Aut(G)$. Otherwise, $G$ would be [SIN$]\_B$ and in this case, $K\_B$ is the trivial group of the identity.


[1] Grosser, Siegfried; Moskowitz, Martin; Compactness conditions in topological groups. J. Reine Angew. Math. 246 1971 1–40, DOI: [10.1515/crll.1971.246.1](https://doi.org/10.1515/crll.1971.246.1).



An [IN$]_B$ group with a non-normal compact $B$-invariant subgroup
Take $G=\text{GL}\_2(\mathbb{Z})\ltimes (\mathbb{R}/\mathbb{Z})^2$ and take $B<G$ to be the $\begin{bmatrix} 1 & \* \\ 0 & 1 \end{bmatrix} \simeq \mathbb{Z}$. Then $K\_B$ is one of the factors $\mathbb{R}/\mathbb{Z}$.


