Alan, who is the brother-in-law of Lilly, is the son-in-law of Sam. June is the mother of Amy, who is the only sister of Lilly. How is Sam related to Lilly?

Let's break down the relations step by step:
1. June is the mother of Amy, and Amy is the only sister of Lilly. This means Amy and Lilly are sisters, and June is the mother of both Amy and Lilly.
2. Alan is the brother-in-law of Lilly and the son-in-law of Sam. Since Amy is Lilly's only sister, Alan must be married to Amy.
3. Since Alan is married to Amy, Alan's father-in-law/mother-in-law is Amy's parent. We are given Alan is the son-in-law of Sam, which means Sam is a parent of Amy.
4. Since Amy and Lilly are sisters, Sam (being Amy's parent) must also be Lilly's father (as June is already established as the mother).

Thus, Sam is the father of Lilly.