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 $G^{0}=O(n)$?