= Solution
Let $P$ be a Sylow p-subgroup. Since $[G:P]$ is invertible in $k$, every $kG$-module is relatively $P$-projective.
Suppose first that $kP$ has only finitely many indecomposable modules $U_1,\ldots,U_t$. For every indecomposable $kG$-module $M$, decompose $\operatorname{Res}_P^GM$ into the $U_i$. Relative projectivity makes $M$ a summand of the corresponding finite direct sum of the $\operatorname{Ind}_P^GU_i$. The <Krull–Schmidt theorem> leaves only finitely many possible indecomposable summands, so $kG$ has finite representation type.
Conversely, suppose $kG$ has finitely many indecomposables. For an indecomposable $kP$-module $U$, the identity double coset in the <Mackey restriction formula> shows that $U$ is a direct summand of
$$
\operatorname{Res}_P^G\operatorname{Ind}_P^GU.
$$
Decomposing the induced module into the finitely many $kG$-indecomposables and restricting them shows, again by Krull–Schmidt, that only finitely many $U$ can occur. Thus
$$
\boxed{kG\text{ has finite representation type}
\Longleftrightarrow kP\text{ has finite representation type}.}
$$
If $P$ is cyclic, the <indecomposable modules of a cyclic p-group in characteristic p> form a finite list. If $P$ is noncyclic, its <Frattini quotient> has rank at least two and therefore has a quotient $C_p\times C_p$. Inflation preserves indecomposability and nonisomorphism, while $k(C_p\times C_p)$ has infinitely many indecomposable modules. The <Higman criterion for finite representation type of a group algebra> now gives
$$
\boxed{kG\text{ has finite representation type}
\Longleftrightarrow P\text{ is cyclic}.}
$$
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