Under what conditions can we use change of variables in an integral with respect to the Hausdorff measure?

Question

Let ${\cal H}^m$ be the $m$-dimensional Hausdorff measure, $A$ be a ${\cal H}^m$-measurable subset of $\mathbb{R}^n$ ($m<n$), and $f:A\rightarrow \mathbb{R}^n$ be an injective Lipschitz function. Under what conditions is it true that

$$ \int_{f(A)}gd{\cal H}^m=\int_{A}g\circ fJfd{\cal H}^m $$

for some function $g:\mathbb{R}^n\rightarrow \mathbb{R}$, where $Jf$ is the Jacobian of $f$?

I know from this post that this does not hold in general, but is there any set of conditions under which this holds?