If h is a path in X starting at x0 and ending at x1 then we
have deﬁned an isomorphism βh : π1 (X, x1 ) → π1 (X, x0 ).
Now suppose that x0 = x1 .
In that case βh : π1 (X, x0 ) → π1 (X, x0 ).
In other words βh is an automorphism of π1 (X, x0 ). Can you describe this
automorphism more concretely?