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Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.

Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding small category, $X$ and $A$ plays the role of morphisms and objects, respectively.

Edit 1: Is there a theory which investigate such type of problems for $A$ not necessarily a single point

In particular, is there a non single retract $A$ of $X=S^{3}$ or $X=S^{7}$ such that $(X,A)$ does not admit a topological groupoid structure.

In this question we do not require that the retracting map has any relation to the source and range maps

Edit 2: As another particular case, is there a groupid structure on $(G,G^{0})$ where $G=Gl(n,\mathbb{R})$ and $G^{0}=O(n)$?

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.

Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding small category, $X$ and $A$ plays the role of morphisms and objects, respectively.

Edit 1: Is there a theory which investigate such type of problems for $A$ not necessarily a single point

In particular, is there a non single retract $A$ of $X=S^{3}$ or $X=S^{7}$ such that $(X,A)$ does not admit a topological groupoid structure.

In this question we do not require that the retracting map has any relation to the source and range maps

Edit 2: As another particular case, is there a groupid structure on $(G,G^{0})$ where $G=Gl(n,\mathbb{R})$ and $G^{0}=O(n)$? That is: are there range, source and composition maps which are continuous which makes $(G,G^{0})$ a groupoid.

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.

Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding small category, $X$ and $A$ plays the role of morphisms and objects, respectively.

Edit: Is there a theory which investigate such type of problems for $A$ not necessarily a single point

In particular, is there a non single retract $A$ of $X=S^{3}$ or $X=S^{7}$ such that $(X,A)$ does not admit a topological groupoid structure.

In this question we do not require that the retracting map has any relation to the source and range maps

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.

Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding small category, $X$ and $A$ plays the role of morphisms and objects, respectively.

Edit 1: Is there a theory which investigate such type of problems for $A$ not necessarily a single point

In particular, is there a non single retract $A$ of $X=S^{3}$ or $X=S^{7}$ such that $(X,A)$ does not admit a topological groupoid structure.

In this question we do not require that the retracting map has any relation to the source and range maps

Edit 2: As another particular case, is there a groupid structure on $(G,G^{0})$ where $G=Gl(n,\mathbb{R})$ and $G^{0}=O(n)$?