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.
- One can easily note that the group $B$ cannot include all the inner automorphisms.
- 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.
 Grosser, Siegfried; Moskowitz, Martin; Compactness conditions in topological groups. J. Reine Angew. Math. 246 1971 1–40, DOI: 10.1515/crll.1971.246.1.