I am reading the article Homotopy stable classification of $BG^{\wedge}_p$ by Martino-Priddy. Let $P_u$, $P_v$ $p$-subgroups of a finite group G, such that $P_u\leq x^{-1}P_v x$ for some $x\in G$, if $X$ is a common indecomposable stable summand of $BP_u$ and $BP_v$, $\iota: X\rightarrow BP_u$ is an inclusion map of $X$ as a summand of $BP_u$, the proof proposition 3.2 part b) seems to say implicitly that the composite $$X\xrightarrow{\iota} BP_u\xrightarrow{Bi\circ Bc_x} BP_v$$ is an inclusion map of X as a summand of $BP_v$. Is it true?.

P.D: All objects here are $p$-completed spectra, X is also a summand of $BG$.