What is the problem with having a hash to group function where you can find a discrete log relation between 2 different hashes?

I was reading some notes on a naive hash to a group function.

Consider a cryptographic Hash function $$H: \{0,1\}^{*}\to \{0,1\}^{k}$$

Consider a Discrete Log Hard Group $G$ with a generator $g$. We can build a Hash to group function $$HG(a) = g^{H(a)}$$

(We raise $g$ to the numerical representation of the Hash output)

Apparently, the problem with this is that it's easy to find a relation between 2 hash outputs of this Hash to the group function.

Assume two inputs, $a_1 \& a_2$

Let numerical representation of $H(a_ 1)$ in $Z_n$ be $n_1$

Let numerical representation of $H(a_2)$ be $Z_n$ be $n_2$

$HG(a_1) = g^{n_1}$

$HG(a_2) = g^{n_2}$

Let ${n_1}^{-1}$ be inverse of $n_1$ in $Z_n$.

Let $c = n_2 \cdot {n_1}^{-1}$

Now ${HG(a_1)}^c = {g^{n_1}}^c = g^{n_1\cdot {n_1}^{-1}.n2} = g^{n_2} = HG(a_2)$

So we can now find a relation between Hash to Group of $a_1$ & $a_2$.

$HG(a_2) = {HG(a_1)}^c$

I understood up to this. However what is the problem if you can find a relationship like this? Can this be used to attack the Hash to Group function in some way?