I interpret the question as whether induced representation of unitarizable representation are unitarizable.

This is true, when $G$ is a locally compact group and $H$ a closed subgroup. It's due to Mackey. The construction involves a (quasi invariant) measure on $H \backslash G$, which behaves nicely under right translation.

For the measure: The existence of the measure is shown in Folland "Harmonic analysis", but he focuses soon thereafter on the compact case. I do not know of any other textbook, which has reference with proofs for this.

For the induced representation: Barut and Raczka "The theory of group representation ... " do quite a good job in the general case, but omit many proofs and have a cumbersome notation (adopted from Mackey's paper).

At a beginner's level? For compact groups, there are more reference. But I think that there a not so many sources, which explain Mackey's theory in general, and none at a beginners level. I suggest to learn a lot of finite/compact group stuff, before going to the more general settings, and consider induction and restriction as functor.

Attention: There are also non-unitarizable representations of $H$, which become unitarizabile after induction. The complementary series representations are examples.

For a finite group and there representation on characteristic zero fields, you can also write down the canonical inner product yourself, its
$$\langle f(g), h(g) \rangle_G = \sum\limits_{H \backslash G} \langle f(g), h(g) \rangle_H,$$
and the well-behavedness in your question with respect to the action makes it well-defined. Perhaps you have learnt the tensor approach first, where this choice is not so obvious, but for function on $G$, this choice is obvious.

In characteristic $p$ the inner product can of course be non-degenerate for obvious reasons.