Can I determine a $p$-group by the number of subgroups

Question

Let $G$ and $H$ be finite $p$-groups of order $p^n$ where $p$ is a prime. Assume that for every $1\leq i\leq n$, the number of subgroups of $G$ and $H$ of order $p^i$ are the same. Can I deduce that $G$ and $H$ are isomorphic?

It seems like it works for $n=1,2,3$ but these cases are more or less trivial. For $n>3$, examples become too complicated for me to investigate.

Answer 1

No of course you cannot deduce that? Why would expect to? (Counterexample: $\mathtt{SmallGroup}(16,i)$ for $i=2$ and $4$; or for $i=5$ and $6$.)