An easy example giving a negative answer. Let $p>2$, $G$ be abelian of type $(p^2,p)$ and $H$ is nonabelian metacyclic of order $p^3$.

$G$ satisfies the above condition iff any its Sylow subgroup is cyclic or generalized quaternion. Such $G$ are described in know papers of H. Zassenhaus and M. Suzuki. There are among them nonsolvable groups, for example, SL(2,5).

Why -1? I gave a correct example.

Let a Sylow p-subgroup is not normal in a p-group G, P and P_1 are distinct Sylow p-subgroups of G. If M is the normalizer of P_1 in G, then |M\cap P|_p<|P|

@JSpecter You have obtained a very weak result in so complicated way. For an easier proof, see any textbook containing sections devoted finite p-groups..

To Strickland. Please, let us know for which $k$ the groups of coclass are classified. Also, for $k>1$ the classification of $p$-groups of class $k$ is absent (it is known that the most $p$-groups have class $2$).

The following result is due to Burnside-Brauer. Let a faithful character $\chi$ of a group $G$ takes exactly $N$ values. Then the sum $1_G+\chi+\dots+\chi^{N-1}$ contains all irreducible characters of $G$. The following dual result is due to S. Garrison: Let $K$ be a $G$-class and $\langle K\rangle=G$, $k$ the corresponding class sum and $m$ is the number of distinct values of the function $\chi\to\frac{\chi(k)}{\chi(1)}$, where $\chi\in\text{Irr}(G)$. Then every element of $G$ can be written as a product of less than $m$ elements of $K$.

A comprehensive consideration of faithful representation one can fiund in Chapter 9. of the book Berkovich-Zhmud;, Character of Finite groups. Part.1. In particular, the following results there proved: (1) (Gasch\"utz): A group $G$ possesses a faithful irreducible character iff its solle is generated by $G$-class. (2) (Zhmud;) The number os kernels of irreducible characters of $G$ is equal to the number of normal closures of elements of $G$.

The result, mentioned in Prof. Robinson's post. is due to F. Klein.