Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites:

 - If we allow the axiom of choice, you can consider the reals as a vector space over the rationals, and get $\frak{c^c}$ homomorphisms. Note that with the axiom of choice this is the same as $\frak c!$(cardinality of set of bijections) and $2^\frak c$.
 - On the other hand there is [the Solovay model](https://en.wikipedia.org/wiki/Solovay_model) which is a model where every set of reals is Baire measurable. By theorem of Pettis all Baire measurable homomorphisms between Polish groups are continuous. There are $\frak c$ continuous homomorphisms $(\mathbb R,+)$ to itself.
 - Asaf Karagila brought up, in a version of this [question on math.se](https://math.stackexchange.com/questions/3367572/can-the-set-of-endomorphisms-of-mathbbr-have-cardinality-strictly-betwe#comment6930439_3367572), that it is consistent that $\aleph_1 \times \frak c$ can not support a group structure.  This means the set of endomorphisms can not have that cardinality(in such models).

**Questions:** Can the cardinality of the set of endomorphisms,$\frak e$, be strictly between $\frak c$ and $\frak{c^c}$? Somewhat vague—is there a sort of classification of what cardinals $\frak e$ could be(for example you need to be able to put a group structure on it, are there any other restrictions)? I am also interested in the same questions except looking at isomorphisms and the cardinals are $\frak c$ and $\frak {c} !$.

(this is an edited version of a question I [asked on math.se](https://math.stackexchange.com/q/3367572/29123), I figured most of the active math.se set theory people already saw it there and I probably should have asked it here anyways)