Homomorphic hash from prime order group $G$ to $Z_p$

Let $G$ be a cyclic group with the generator $g$ and of prime order $p$ such that the discrete-logarithm problem is hard in $G$.

A hash function is homomorphic if $H(a\ast b)=H(a)\cdot H(b)$ (where the operations $\ast$ and $\cdot$ depend on the groups). Here we do not expect the hash function to be compressing, but collision-resistance (CR) and efficiently computeable.

Now the question is, if there exist such homomorphic hash function from group $G$ to $Z^+_p$?

Yes. The function is usually referred to as the discrete logarithm function. It is defined by $$H:G\to(\mathbb Z/p\mathbb Z)^+$$ $$H(g^X)=X$$

The function always exists, but if $G$ is a cryptographic group, then $H$ should be infeasible to compute. Technically, there is one such function for every $g$, but they are all multiples of each other.

We would typically just call this a function rather than a hash function. It's certainly not a cryptographic hash function as it can be inverted with $O(\log p)$ operations in $G$.

ETA: Note that by the homomorphic property $H(h^a)=aH(h)$ and so the value of $H(g)$ completely determines the function. In other words the discrete logarithm function and its multiples represent all possible homomorphic functions from $G$ to $(\mathbb Z/p\mathbb Z)^+$. There are no others.