# Homomorphism from automorphism group of connected groupoid to outer automorphism group at a point

Consider a connected groupoid with $X$ its collection of points and $P(x,y)$ denoting the paths between points $x,y \in X$. Let $x \in X$ be a point. Denote by $A$ the group of automorphisms of the groupoid $X$, and by $G_x$ the group $P(x,x)$ under composition. Then, there is a canonical homomorphism:
$A \to \operatorname{Out}(G_x)$
• Given $g \in A$, pick a path from $x$ to $y = g.x$. Call this path $p$.
• Consider now the map sending $h \in G_x$ to $p^{-1} \circ (g.h) \circ p$. Note that this map is an automorphism of the group $G_x$.
• Now, the automorphism we got depended on the choice of path from $x$ to $y$. Choosing a different path gives an automorphism that differs from the original one by an inner automorphism. Hence, we get a well-defined element of $\operatorname{Out}(G_x)$.