By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group
F
F
. We consider individual comultiplications on
F
F
and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of
F
F
. For a comultiplication
m
m
of
F
F
we define a subset
Δ
m
⊆
F
\Delta _{m} \subseteq F
of quasi-diagonal elements which is basic to our investigation of associativity. The subset
Δ
m
\Delta _{m}
can be determined algorithmically and contains the set of diagonal elements
D
m
D_{m}
. We show that
D
m
D_{m}
is a basis for the largest subgroup
A
m
A_{m}
of
F
F
on which
m
m
is associative and that
A
m
A_{m}
is a free factor of
F
F
. We also give necessary and sufficient conditions for a comultiplication
m
m
on
F
F
to be a coloop in terms of the Fox derivatives of
m
m
with respect to a basis of
F
F
. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group
Aut
F
\operatorname {Aut} F
on the set of comultiplications of
F
F
. We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.