Extension of a real valued function preserving given properties

Question

Consider a function defined on a closed interval $f:[-a,a]\subseteq\mathbb{R}\longrightarrow\mathbb{R}$ for some $a>0$. Suppose that:

1. $f$ is of class $C^2$ on $[-a,a]$,

2. there is $c>0$ such that $|f'(x)|\geq c$, $\forall x\in [-a,a]$.

Is there an explicit extension $g:\mathbb{R}\longrightarrow\mathbb{R}$ such that:

A) $g(x)=f(x)$, $\forall x\in[-a,a]$,

B) $g$ satisfies properties 1. and 2. on the whole $\mathbb{R}$,

C) $g$ and $g'$ are globally Lipschitz.