What is a (simple) example of a group $G$ with an endomorphism $G\rightarrow G$ That is surjective, but not injective?
There are no examples with $G$ finite. So you need $G$ to be infinite.
Consider the set of all polynomials with rational coefficients as an additive group.
Then the derivation map $p\mapsto p'$ is a surjective, but not injective, endomorphism.
Or consider the set of complex numbers of modulus $1$ as a multiplicative group.
Then the square map $z\mapsto z^2$ is a surjective, but not injective, endomorphism.