A comparison between the fundamental groupoid and the fundamental group

Question

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to the fundamental group of $Y$ ?

I guess that there exists such a pair of topological spaces. I don't know an example though. I am very interested to see such a pair.

Edit: The first version of the question was already solved by Zev Chonoles. Here is the second version of the question

Are there two path-connected topological spaces $X,Y$ such that:

1) $|X|=|Y|$

2) The fundamental group of $X$ is isomorphic to the fundamental group of $Y$

3) The fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$

In other words, this is a comparison between the fundamental groupoid functor and the combined use of the fundamental group and the forgetful functor from Top to Set

Thank you

Answer 1

It suffices to prove the following:

If $X$ and $Y$ are connected groupoids, then given any bijection $F : \operatorname{ob} X \to \operatorname{ob} Y$ and any group isomorphism $\Phi : X(x, x) \to Y(y, y)$ (where $x$ is in $X$ and $y = F x$), there is a groupoid isomorphism $X \to Y$ extending the given data.

Choose an isomorphism $p_{x'} : x \to x'$ for each $x'$ in $X$, with $p_x = \mathrm{id}_x$, and choose an isomorphism $q_{y'} : y \to y'$ for each $y'$ in $Y$, with $q_y = \mathrm{id}_y$. We define a functor $F : X \to Y$ as follows: $F$ acts on objects as the given bijection $\operatorname{ob} X \to \operatorname{ob} Y$, and for each morphism $f : x' \to x''$ in $X$, we define $$F f = q_{y''} \circ \Phi (p_{x''}^{-1} \circ f \circ p_{x'}) \circ q_{y'}^{-1}$$ It is easily shown that $F$ is indeed a functor and has the desired properties.

Answer 2

One example would be with $X$ and $Y$ indiscrete spaces of cardinalities $1$ and $2$ respectively. Their fundamental groupoids have different cardinalities and therefore are not isomorphic, but both of their fundamental groups are trivial.