I'm wondering about the following 2 generalizations of Cayley's Theorem (every group embeds in a symmetric group). If these are known to be true/false, references would be appreciated.
1) (Weak Version) Given any finite collection of (not necessarily distinct) finite groups, can we embed them simultaneously in a (finite) group so that they have pairwise-disjoint intersection (i.e., intersect only at the identity)?
2) (Strong Version) Given an arbitrary set of arbitrary groups, can we embed them simultaneously in some group so that they have pairwise-disjoint intersection? It seems like this version may run into set-theoretic difficulties; if so an explanation/reference for those would also be welcome.
Note of course that if such a group exists for a given collection, then we can embed it in a symmetric group by Cayley's Theorem (so these are indeed generalizations).
Edit: To make it more interesting / rule out the obvious answer pointed out in the comments, can we arrange the embeddings so that the groups' normalizers are pairwise disjoint?