Approximation of Lipschitz curves by $C^{1,1}$ curves

Question

Given a curve $\gamma :[0,1] \to \mathbb{R}^2 $ define the set $F(\gamma)=\cup_{t\in [0,1]} B_1(\gamma(t))$.

Given a Lipschitz curve $\gamma$, does it exist a curve $\sigma :[0,1]\to \mathbb{R}^2$ of class $C^{1,1}$ with curvature bounded by $1$ such that $\sigma(0)=\gamma(0)$, $\sigma(1)=\gamma(1)$ and $F(\sigma) \subset F(\gamma)$?