Subnormalizer subset

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. The subnormalizer subset (sometimes termed the subnormalizer) of $$H$$ in $$G$$ is defined as:

$$\{ g \in G \mid H \triangleleft \triangleleft \langle H, g \rangle \}$$.

The term subnormalizer is sometimes reserved for the situation where this subset is a subgroup and $$H$$ is subnormal in it, in which case the original subgroup $$H$$ is termed a subgroup having a subnormalizer. Note that the subnormalizer subset need not be a subgroup because subnormality is not upper join-closed.