I was recently reading up on Fraïssé limits in Hodges' "A shorter model theory." I was trying to think of some examples and wanted to see if I could take the Fraïssé limit on the category of finite groups. It is clear that this category has the hereditary and joint embedding property. However, I am having trouble showing that this category has the amalgamation property, by which I mean that if there are finite groups $G, H, K$ such that there are embeddings $\varphi: G\hookrightarrow H$ and $\psi: G\hookrightarrow K$, then there exists a finite group $J$ and embeddings $\vartheta: H\hookrightarrow J$ and $\eta: K\hookrightarrow J$ such that $\vartheta\circ \varphi = \eta\circ \psi$.
One simplification is that we can, by taking isomorphic copies if necessary, assume that $H\cap K = G$, and that $\varphi $ and $\psi$ are just inclusion maps. I am not even sure what group $J$ should be. Any help would be appreciated.