Let $G$ be a group with $H \leq K \leq G$. Let $\chi \in \text{Irr}(G)$, $\psi \in \text{Irr}(K)$ and $\phi \in \text{Irr}(H)$.
Suppose that $\phi$ is an irreducible constituent of $\psi_H$ i.e. $ \langle \psi_H, \phi \rangle \neq 0$ and that $\psi$ is an irreducible constituent of $\chi_K$, i.e. $\langle \chi_K, \psi \rangle \neq 0$. I want to show that $\phi$ is an irreducible constituent of $\chi_H$ i.e. to show that $\langle \chi_H, \phi \rangle \neq 0$
Here's my attempt so far: By Frobenius Reciprocity, $\langle \psi_H, \phi \rangle = \langle \psi, \phi^K \rangle$. Using the property of trasitivity of induction, we get that $\langle \psi, \phi^K \rangle = \langle \psi^G, (\phi^K)^G \rangle =\langle \psi^G, \phi^G \rangle \neq 0 $.
Furthermore, since $\langle \chi_K, \psi \rangle \neq 0$, we have that $\langle \chi, \psi^G \rangle \neq 0$.
I need to show $\langle \chi_H, \phi \rangle \neq 0$. Equivalently, $ \langle \chi, \phi^G \rangle \neq 0$. I can 'see' the result from the above workings but I am unable to conclude substantially.
I suppose you know that if $\psi$ is an irreducible constituent of two representations $\rho,\sigma$, then $\langle\rho,\sigma\rangle\neq0$. Now apply that to $\rho=\chi_K$ et $\sigma=\phi^K$, and apply the Frobenius reciprocity of your choice, and transitivity of restriction or induction.
But the way, the first equality you mention in the question after "trasitivity of induction" is not valid.