If $A<S_\mathbb{N},$ are there always $B,C<S_\mathbb{N}$ with $B\cap C=A?$

Question

Let A be a proper (non maximal) subgroup of the countable symmetric group $S_\mathbb{N}.$ In general, can we always find $B,C$ proper subgroups of $S_\mathbb{N}$ with $A<B$ and $A<C$ such that $B\cap C=A?$

I found such an A with this property and I want to know if this makes it special or if every such A has this property.