For a local field $F$ and an Artinian local coefficient ring $\Lambda$ with the same positive residue characteristic $p$ we define, for any $e\in {\mathbb N}$, a category ${\mathfrak C}^{(e)}(\Lambda )$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $\Lambda$ to ${\mathfrak C}^{(e)}(\Lambda )$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(\Lambda )$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $\Lambda$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${\mathfrak C}^{(e)}(\Lambda )$ with trivial central character is equivalent to a category of representations of a certain subgroup of ${\rm Aut}(X)$ consisting of ‘locally algebraic automorphisms of level $e$'. For $e=1$ there is a functor from this category to that of modules over the (usual) pro-$p$-Iwahori Hecke algebra ; it is a bijection between irreducible objects. Finally, we present a parallel of Colmez' functor $V\mapsto {\bf D}(V)$ : to objects in ${\mathfrak C}^{(e)}(\Lambda )$ (for any $F$) we assign certain étale $(\varphi ,\Gamma )$-modules over an Iwasawa algebra ${\mathfrak o}[[\widehat {N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawa algebra ${\mathfrak o}[[{N}_{0}]]$. This assignment preserves finite generation.