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Zero comodule!!

If you want something more halloweeny, take $G$ to be a field, $H$ - a Hopf algebra, not a field, $V$ - any simple nontrivial $H$-comodule...

EDIT: as asked, $G$ is any Hopf algebra with a simple non-trivial comodoule, $H=G\otimes G$, the map is $x\mapsto x \otimes 1$, $V$ is a simple nontrivial $1\otimes G$-comodule (and hence $H$-comodule)... Bingo!

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Zero comodule!!

If you want something more halloweeny, take $G$ to be a field, $H$ - a Hopf algebra, not a field, $V$ - any simple nontrivial $H$-comodule...