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?