Let me first give the definition of a local coefficient system (see also [2, p. 257] and [3, p. 35]):


Let $X$ be a topological space. A local coefficient system is a functor from the category $\Pi\_1(X)$ (= the fundamental groupoid) to the category AbGrp of abelian groups. Such a functor assigns to each $x \in X$ an abelian group $G(x)$ and to each homotopy class $\xi \in \pi\_1(X; x\_1, x\_2)$ (the set of all endpoint-preserving homotopy classes of paths from $x\_1$ to $x\_2$) a homomorphism $G(\xi): G(x\_2) \to G(x\_1)$; these are required to satisfy


(i) if $\xi \in \pi\_1(X, x) = \pi\_1(X; x, x)$ is the identity, then $G(\xi): G(x) \to G(x)$ is the identity;


(ii) if $\xi \in \pi\_1(X; x\_1, x\_2)$, $\eta \in \pi\_1(X; x\_2, x\_3)$, then $G(\xi \eta) = G(\xi) \circ G(\eta): G(x\_3) \to G(x\_1)$


So my question is:


It is easy to see that every bundle of groups (defined in [1, p. 330]) is a local coefficient system, but I think the converse is not true (as stated without proof in [3, p. 35]), so I am looking for a (non-trivial) local coefficient system which is not a bundle of groups. 


**Please note:** A good choice for the space $X$ in the above definition of a local coefficient system could for example be the Hawaiian earrings.


Thank you in advance!


References:


[1] Hatcher, Allen: Algebraic Topology, 2002, <https://www.math.cornell.edu/~hatcher/AT/AT.pdf>, p. 330 


[2] Whitehead, George: Elements of Homotopy Theory, New York: Springer, 1978 (Graduate Texts in Mathematics Vol. 61), p. 257


[3] Hutchings, Michael: Introduction to higher homotopy groups and obstruction theory, 2011, <https://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf>, p. 35



(Non-trivial) local coefficient system which is not a bundle of groups
Strip away the group structure and you get the simpler question: Does every functor from the fundamental groupoid of $X$ to Set correspond to a covering space (a bundle of sets)? As a special case this includes the question, does every subgroup of the fundamental group of a path-connected space come from a connected covering space.The answer to all of these is no. A standard example is when $X$ is the "Hawaiian earring" a.k.a. "clamshell space" a.k.a. "shrinking wedge of circles" (union of infinite sequence of circles in the plane with diameters tending to zero and one point in common).


I like to organize these ideas like this: There is a functor, obviously faithful, from the category $Cov(X)$ of covering spaces of $X$ to the category $Fun(\Pi\_1(X),Set)$. It is not always fully faithful, but it is if $X$ is locally path-connected. In this case it is not always an equivalence of categories, but it is if $X$ is, as they say, "semi-locally simply connected". 


