First agemo subgroup need not have every element a pth power

Statement
For any prime number $$p$$, it is possible to have a finite $$p -group P (hence, P is a group of prime power order with the property that the first agemo subgroup of P, defined as:

contains an element g for which there is no h satisfying g = h^p$$.

Case $$p = 2$$
Consider the group SmallGroup(16,3), given by the presentation:

$$G := \langle a,b,c \mid a^4 = b^2 = c^2 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle$$

Then, the set of squares in $$G$$ is $$\{ e, a^2, a^2b \}$$. The subgroup generated by these is $$\{ e, a^2, a^2b, b \}$$, which contains the additional element $$b$$ that is not a square.

Another example is SmallGroup(16,4), given by the presentation:

$$G := \langle a,b,c \mid a^2 = b^4 = e, b^2 = c^2, ab = ba, ac = ca, cbc^{-1} = ab \rangle$$

The set of squares in $$G$$ is $$ \{ e, a, b^2 \}$$. The subgroup generated by these is $$\{ e, a, b^2, ab^2 \}$$, which contains the additional element $$ab^2$$ which is not a square.