TITLE: A (nameless?) product in the category of groups, and its properties
QUESTION [7 upvotes]: Suppose I have a group $G,$ with normal subgroups $K, L$ such that $G=KL$ but $K\cap L \neq \{1\}.$ If the intersection were trivial, we would say that $G$ is the direct product of $K$ and $L,$ but this operation to direct product is like free product with amalgamation is to free product. Any names for it out there?
I am really more interested in the properties of this more than the name (though the latter might help me find the former) -- for example, given three groups $H_1, H_2, H3,$ is there a way to classify $G$ where $K\simeq H_1, L\simeq H_2, K\cap L \simeq H_3?$ What if $H_1, H_2$ are finitely generated free groups, and $H_3$ is an infinitely generated free group?

REPLY [2 votes]: As already mentioned above, the group in question is an extension
$$1\rightarrow K\cap L \rightarrow KL \rightarrow K/K\cap L \times L/K\cap L \rightarrow 1,$$
where we abbreviate the rightmost (a priori) non trivial group by $H$.
In particular $K\cap L$ is normal in $KL=G$, so $G$ is isomorphic to a subgroup of the wreath product $K\cap L \wr H$. I don't know if this point of view is helpful, but I for myself would try to use wreath products here.