Restriction and induction along a subgroup preserve projective modules

Let $P$ be a projective $kG$-module. By [F1], it is a direct summand of a free module $(kG{)}^n$. Restricting to $H$ preserves direct sums and summands. By [L2] with $W=kH$, the restricted regular module ${\mathrm{Res}}_H^G(kG)$ is a finite direct sum of copies of $kH$, hence is free as a $kH$-module. Therefore ${\mathrm{Res}}_H^{G}P$ is a direct summand of a free $kH$-module and is projective by [F1].

Let $Q$ be a projective $kH$-module. To prove that ${\mathrm{Ind}}_H^{G}Q$ is projective, use the lifting characterization in [F1]. Given a surjection $u:X\to Y$ of $kG$-modules and a map $\varphi :{\mathrm{Ind}}_H^{G}Q\to Y$, adjunction [L1] turns $\varphi$ into a map $\psi :Q\to {\mathrm{Res}}_H^{G}Y$. The restriction of $u$ is still surjective, so projectivity of $Q$ lifts $\psi$ to $\tilde{\psi }:Q\to {\mathrm{Res}}_H^{G}X$. Applying [L1] again yields a lift $\tilde{\varphi }:{\mathrm{Ind}}_H^{G}Q\to X$ of $\varphi$. Thus ${\mathrm{Ind}}_H^{G}Q$ is projective.

Steps 1.1 and 2.1 prove that both restriction and induction preserve projectives.