No, an easy example is when $G= Z/2Z {\bf Z}/2{\bf Z} \ltimes (Z/pZ)^2 {\bf Z}/p{\bf Z})^2$ where the action is by switching the factors and $H = (Z/pZ)^2. G {\bf Z}/p{\bf Z})^2$.
$G$ has 2p $2p$ 1dimrepresentations and (p^2-p)/2 $(p^2-p)/2$ 2dim representations which for big p-es $p$ is smaller than the numer number of representations for H $H$ which is p^2$p^2$.