Linear bilipschitz extension property

We give a sufficient geometric condition for a subset $A$ of $\mathbb{R}^n$ to enjoy the following property for a fixed $C\geqslant1$ There is $\delta>0$ such that for $0\leqslant\varepsilon\leqslant\delta$, each $(1+\varepsilon)$-bilipschitz map $f\colon A\to\mathbb{R}^n$ extends to a $(1+C\varepsilon)$-bilipschitz map $F\colon\mathbb{R}^n\to\mathbb{R}^n$.