Hash function and operation commuting over composition

Is there a hash function $H$ and operation $\otimes$, which fulfill the following property?

$$ H(A) \otimes H(B) = H(A \otimes B) $$

$A$ and $B$ are two byte blocks of identical length, if necessary restricted to a fixed length (e.g. 128 bytes). $H$ should be a cryptographic hash function, in particular, it should be pre-image and collision resistant.

Idea

One idea based on a system of equations using $XOR$ I asked in the mathematics forum. But I am mostly interested in existing approaches or a reasoning why this might not be possible.

If you allow different operations $(\oplus, \otimes)$ on the input and the output, then there is such a property for the Pedersen hash function. Fix a group $\mathbb{G}$ of order $p$ with two generators $(g,h)$, and assume that computing the discrete logarithm of $h$ in base $g$ is hard. Then the function $H: \mathbb{Z}_p \times \mathbb{Z}_p \mapsto \mathbb{G}$ which maps $(a,b) \in \mathbb{Z}_p \times \mathbb{Z}_p$ to $H(a||b) = g^ah^b$ is a collision-resistant hash function (under the discrete logarithm assumption) which is compressing by roughly a factor 2 (over groups where group elements can be represented compactly).

Then, defining $\oplus: ((a,b), (a',b')) \rightarrow (a+a', b+b')$ and $\otimes$ to be the group operation, we have $H(a,b)\otimes H(a',b') = H((a,b)\oplus (a',b'))$.

I am not aware of any example where $\oplus = \otimes$.

Then