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Let $G$ be a group and $H$ a subgroup of $G$.
Which of the following properties does not always imply that $H$ is the kernel of some group homomorphism of $G$?
$[G:H]=3$ and $G$ has even order.
$[G:H]=3$ and $G$ has odd order divisible by $3$.
$|G|=35$ and $[G:H]=5$