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Peter probably wouldn’t have a pet, but, if he did, he’d get a gecko or something small and unassuming, I think.
James would own a dog— a husky, probably, something he can have fun and workout with.
Remus would have a cat, probably a stray with some kind of disability he could just sit with and read with some tea (and water, for the kitten).
Sirius would definitely get a massive dog, a rescue, I’d say. Some kind of huge, scary mutt (like Padfoot) with a bad history, maybe of violence or acting out due to abuse.