  For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ -
element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag
poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$.
We say that a transformation $\alpha \in I_n$ is \textit{fence-preserving} if
$x <_f y$ implies that $x\alpha <_f y\alpha$, for all $x, y$ in the domain of
$\alpha$. In this paper, we study the semigroup $PFI_n$ of all partial
fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{\alpha \in
PFI_n : \alpha^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and
contains all regular elements of $PFI_n.$ We characterize the Green's relations
for the semigroup $IF_n$. Further, we prove that the semigroup $IF_n$ is
generated by its elements with \rank$\geq n-2$. Moreover, for $n \in
2\mathbb{N}$ we find the least generating set and calculate the rank of $IF_n$.
