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In my interpretation of your description, a sum-of-induceds is just the permutation representation of some (not necessarily connected) G-set. The representation $\mathrm{Hom}(V_1,V_2)$ for two permutation representations is itself permutation: it's the permutation action on product of the G-sets.

So, $\mathrm{Ext}^1(V_1,V_2)=H^1(G,\mathrm{Hom}(V_1,V_2))$ which you claim vanishes. So it sounds to me like every extension between permutation representations splits. [EDIT: I'm no longer nervous] I'm still a little nervous about this cohomology vanishing, but I don't know integral representation theory so well, so it's hard for me to say.

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In my interpretation of your description, a sum-of-induceds is just the permutation representation of some (not necessarily connected) G-set. The representation $\mathrm{Hom}(V_1,V_2)$ for two permutation representations is itself permutation: it's the permutation action on product of the G-sets.

So, $\mathrm{Ext}^1(V_1,V_2)=H^1(G,\mathrm{Hom}(V_1,V_2))$ which you claim vanishes. So it sounds to me like every extension between permutation representations splits. I'm still a little nervous about this cohomology vanishing, but I don't know integral representation theory so well, so it's hard for me to say.