Let $f:G\to G$ be a surjection from a torsion group $G$ onto itself.
Let the kernel have infinite cardinality: $\lvert\ker(f)\rvert=\aleph_0$
What category of function on groups is this?
To my mind this cannot be a group automorphism because at the very least an automorphism must map the identity onto itself in order to satisfy $f(a\cdot b)=f(a)\circ f(b)$.
Clearly $f$ does not yield distinct inverses.
Is $f$ therefore an endomorphism instead?
In part, I'm asking whether an endomorphism on a torsion group can have a kernel with infinite cardinality - what with the elements themselves having finite order and all.
I'm asking with half an eye on groups of intermediate growth such as the Grigorchuk group, and any torsion group structure on which variants of the Collatz function might be an endomorphism.
I don't think there is an accepted name to those types of functions except for saying what they are: surjective homomorphism with some specified size of kernel. You can say surjective endomorphism, or epic endomorphsims(which sounds sort of cool). The endomorphism means it is a morphism between the same object. In the category of groups surjective morphisms and epimorphism/epic morphism are the same(in general they are nor the same thing).
You mention your motiviation is studying torsion groups of intermediate growth(so finitely generated) with a function like the above. As far as I can tell, it is not known if such torsion f.g. groups exist even without the condition of intermediate growth. The Grigorchuck group does not have any homomorphism like that since it is just infinite which is an infinite group where all proper quotients are finite.
Without finitely generated condition there are groups like that, for example $ \bigoplus_{i \in \mathbb N} \mathbb Z / 2 \mathbb Z$.