 Solution

False. Let $G=S_3$ and $H=K=⟨(12)⟩$. For $g=(13)$, direct multiplication gives the double cosetIt has four elements, and $4\nmid 6=|S_3|$. This is a concrete example of double-coset sizes not dividing group order; Lagrange's theorem applies to subgroups, not to arbitrary double cosets.