For finite groups and a field $F$ of characteristic zero, you can identify the $G$-endomorphisms
of two induced representations
$$ Hom_G( I_H^G \pi, I_K^G \sigma) $$
with the space of functions $f: H \backslash G / K \rightarrow Hom_F(V_\pi, V_\sigma*)$ with $f(hgk)= \pi(h) f(g) \sigma(k)$.
If $H=G$ and $\pi = \sigma$, i.e., the case you are interested in, everything becomes a convolution algebra. If you find a basis, they give you projections to irreducible components, and you can decompose explicitly.

Equivalently, you could use Frobenius reciprocity
$$ Hom_H( \pi, R_H I_K^G \sigma) $$
with Mackey induction restriction formula giving you a decomposition
$$R_H I_K^G \sigma$$
in terms of $H\backslash G/K$.

All this is very combinatorial. There is no general algorithm for general $G$ even in the case of complex representation, e.g., nobody knows how to decompose the parabolic induction in $GL_3(\mathbb{Z} / p^N)$. So, one might say decomposing induced representations into irreducibles is an art rather then a theory.

The situation is slightly different in the special case, where $H$ is **normal**. Then induction exhaust all irreducible representations and it is easy to parametrize them. However, also here you encounter the same difficulties only later:( To understand this special case, I suggest you should learn Clifford theory.

*Edit in response to a comment:* Let $N$ be a normal subgroup. $G$ acts on the representation $\pi$ of $N$ via $g: \pi \mapsto \pi^g(x) = \pi(gxg^{-1})$.

$Ind_N^G \pi$ decomposes with single multiplicity iff $End_G( I_N^G \pi)$ is abelian.

$Ind_N^G \pi$ is irreducible, if the cardinality of the $G$ orbit $\pi$ has the same cardinality as $G/N$.

More generally, the cardinality of representations contained in $I_N^G \pi$ is given by $C(\pi) /N$ for the centralizers $C(\pi) = \{ g \in G : \pi^g =\pi\}$.

1 is obvious by Schur's lemma: Assume $I_N^G \pi = \bigoplus \sigma^{\oplus m_\sigma}$ then as algebra isomorphisms
$$ End_G( I_N^G \pi) = \bigoplus End_G( \sigma^{\oplus m_\sigma}) = \bigoplus M(m(\sigma), F).$$

Proof of 2+3 via Mackey formula, Schur's lemma and Frobenius reciprocity: As vectorspace isomorphisms
$$Endo_G(I_N^G \pi) = Hom_N(\pi , Res\, I_N^G \pi) = \bigoplus_{g \in G/N} Hom_N(\pi, \pi^g).$$