TLDR: How is the Galois action on etale path torsors defined?

Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{et}}(\overline{X},\overline{x})$ denote the etale fundamental group of $\overline{X} := X\otimes\overline{k}$ based at the geometric point associated to $x$. Similarly we may consider $\pi_1^{\text{et}}(\overline{X},\overline{y})$, and $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, which denotes the set of etale paths from $\overline{x}$ to $\overline{y}$.

The Grothendieck $\pi_1$-exact sequence shows that there is a natural $G_k$ (absolute Galois) action on $\pi_1^{\text{et}}(\overline{X},\overline{x})$, and $\pi_1^{\text{et}}(\overline{X},\overline{y})$. I read in several places that there is a natural Galois action on $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, does anybody know how it is defined?

My attempt: the way the Galois group acts on $\pi_1^{\text{et}}(X,\overline{x})$, for example, is via conjugation by a loop in $\pi_1^{\text{et}}(X,\overline{x})$, i.e. $g\in G_k, \gamma_x\in \pi_1^{\text{et}}(X,\overline{x})$, then $g(\gamma_x) = \gamma_{g,x}^{-1}\gamma_x\gamma_{g,x}$. Similarly for $\pi_1^{\text{et}}(X,\overline{y})$. So it appears that there is a natural action of $G_k$ on the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, by sending a path $p\in \pi_1^{\text{et}}(X,\overline{x},\overline{y})$ to $\gamma_{g,x}^{-1}p\gamma_{g,y}$. It would be very convenient if this action may be shown to restrict from an automorphism of the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, to the geometric path space $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$. How do I proceed?

Another question, in case anybody is feeling generous, is if the etale path space has a geometric interpretation, similar to how the etale fundamental group has. I am familiar with a Tannakian description but do not find it extremely illuminating.

EDIT: I would be very appreciative of a Tannakian argument, however.

TLDR: How is the Galois action on étale path torsors defined?

Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\overline{x})$ denote the étale fundamental group of $\overline{X} := X\otimes\overline{k}$ based at the geometric point associated to $x$. Similarly we may consider $\pi_1^{\text{ét}}(\overline{X},\overline{y})$, and $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$, which denotes the set of étale paths from $\overline{x}$ to $\overline{y}$.

The Grothendieck $\pi_1$-exact sequence shows that there is a natural $G_k$ (absolute Galois) action on $\pi_1^{\text{ét}}(\overline{X},\overline{x})$, and $\pi_1^{\text{ét}}(\overline{X},\overline{y})$. I read in several places that there is a natural Galois action on $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$, does anybody know how it is defined?

My attempt: the way the Galois group acts on $\pi_1^{\text{ét}}(X,\overline{x})$, for example, is via conjugation by a loop in $\pi_1^{\text{ét}}(X,\overline{x})$, i.e. $g\in G_k, \gamma_x\in \pi_1^{\text{ét}}(X,\overline{x})$, then $g(\gamma_x) = \gamma_{g,x}^{-1}\gamma_x\gamma_{g,x}$. Similarly for $\pi_1^{\text{ét}}(X,\overline{y})$. So it appears that there is a natural action of $G_k$ on the arithmetic path space $\pi_1^{\text{ét}}(X,\overline{x},\overline{y})$, by sending a path $p\in \pi_1^{\text{ét}}(X,\overline{x},\overline{y})$ to $\gamma_{g,x}^{-1}p\gamma_{g,y}$. It would be very convenient if this action may be shown to restrict from an automorphism of the arithmetic path space $\pi_1^{\text{ét}}(X,\overline{x},\overline{y})$, to the geometric path space $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$. How do I proceed?

Another question, in case anybody is feeling generous, is if the étale path space has a geometric interpretation, similar to how the étale fundamental group has. I am familiar with a Tannakian description but do not find it extremely illuminating.

EDIT: I would be very appreciative of a Tannakian argument, however.

TLDR: How is the Galois action on etale path torsors defined?

Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{et}}(\overline{X},\overline{x})$ denote the etale fundamental group of $\overline{X} := X\otimes\overline{k}$ based at the geometric point associated to $x$. Similarly we may consider $\pi_1^{\text{et}}(\overline{X},\overline{y})$, and $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, which denotes the set of etale paths from $\overline{x}$ to $\overline{y}$.

The Grothendieck $\pi_1$-exact sequence shows that there is a natural $G_k$ (absolute Galois) action on $\pi_1^{\text{et}}(\overline{X},\overline{x})$, and $\pi_1^{\text{et}}(\overline{X},\overline{y})$. I read in several places that there is a natural Galois action on $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, does anybody know how it is defined?

My attempt: the way the Galois group acts on $\pi_1^{\text{et}}(X,\overline{x})$, for example, is via conjugation by a loop in $\pi_1^{\text{et}}(X,\overline{x})$, i.e. $g\in G_k, \gamma_x\in \pi_1^{\text{et}}(X,\overline{x})$, then $g(\gamma_x) = \gamma_{g,x}^{-1}\gamma_x\gamma_{g,x}$. Similarly for $\pi_1^{\text{et}}(X,\overline{y})$. So it appears that there is a natural action of $G_k$ on the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, by sending a path $p\in \pi_1^{\text{et}}(X,\overline{x},\overline{y})$ to $\gamma_{g,x}^{-1}p\gamma_{g,y}$. It would be very convenient if this action may be shown to restrict from an automorphism of the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, to the geometric path space $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$. How do I proceed?

Another question, in case anybody is feeling generous, is if the etale path space has a geometric interpretation, similar to how the etale fundamental group has. I am familiar with a Tannakian description but do not find it extremely illuminating.

TLDR: How is the Galois action on etale path torsors defined?

Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{et}}(\overline{X},\overline{x})$ denote the etale fundamental group of $\overline{X} := X\otimes\overline{k}$ based at the geometric point associated to $x$. Similarly we may consider $\pi_1^{\text{et}}(\overline{X},\overline{y})$, and $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, which denotes the set of etale paths from $\overline{x}$ to $\overline{y}$.

The Grothendieck $\pi_1$-exact sequence shows that there is a natural $G_k$ (absolute Galois) action on $\pi_1^{\text{et}}(\overline{X},\overline{x})$, and $\pi_1^{\text{et}}(\overline{X},\overline{y})$. I read in several places that there is a natural Galois action on $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$, does anybody know how it is defined?

My attempt: the way the Galois group acts on $\pi_1^{\text{et}}(X,\overline{x})$, for example, is via conjugation by a loop in $\pi_1^{\text{et}}(X,\overline{x})$, i.e. $g\in G_k, \gamma_x\in \pi_1^{\text{et}}(X,\overline{x})$, then $g(\gamma_x) = \gamma_{g,x}^{-1}\gamma_x\gamma_{g,x}$. Similarly for $\pi_1^{\text{et}}(X,\overline{y})$. So it appears that there is a natural action of $G_k$ on the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, by sending a path $p\in \pi_1^{\text{et}}(X,\overline{x},\overline{y})$ to $\gamma_{g,x}^{-1}p\gamma_{g,y}$. It would be very convenient if this action may be shown to restrict from an automorphism of the arithmetic path space $\pi_1^{\text{et}}(X,\overline{x},\overline{y})$, to the geometric path space $\pi_1^{\text{et}}(\overline{X},\overline{x},\overline{y})$. How do I proceed?

Another question, in case anybody is feeling generous, is if the etale path space has a geometric interpretation, similar to how the etale fundamental group has. I am familiar with a Tannakian description but do not find it extremely illuminating.

EDIT: I would be very appreciative of a Tannakian argument, however.