Small doubling does not control tripling in a noncommutative group

Let $H$ be a finite subgroup and choose $x$ with $H\cap xH{x}^{-1}=\{1\}$. For $A=H\cup \{x\}$, the set $A^2$ has size at most $3|H|+1$, while $A^3$ contains the double coset $HxH$ of size $|H{|}^2$. Thus bounded doubling alone gives no tripling bound in arbitrary groups.