If two sets have the same outer measure, do their images under a function (or linear map) have the same measure?

Question

I'm studying for an exam and thinking about the following question: If $E,E' \in \mathbb{R}^d$ and $m_*(E)=m_*(E')$, is $m_*(f(E))=m_*(f(E'))$, where $f$ is a function or linear map? I don't think it's true for any function $f$ since if $E \cap E'=\emptyset$ and $f(x)=0$ (for $x\in E$) and $f(x)= x$ (for $x \in E^c$), then $m_*(f(E))=m_*(\{0\})=0$ but $m_*(f(E'))=m_*(E')$ is not necessarily $0$. (There's probably a better, more explicit counter-example here.) What about for linear maps $f \colon \mathbb{R}^d \to \mathbb{R}^d$?