Collection of groups of prime-to-the-prime order and prime exponent does not satisfy weak normal replacement condition

Statement
Let $$p$$ be a prime.

Let $$\mathcal{S}$$ be the collection of groups of order $$p^p$$ and exponent $$p$$.

Then, $$\mathcal{S}$$ is not a fact about::collection of groups satisfying a weak normal replacement condition for the prime $$p$$.