Retract not implies free factor in free group

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a free group. That is, it states that in a free group, every subgroup satisfying the first subgroup property (i.e., retract) need not satisfy the second subgroup property (i.e., free factor)
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Statement

It is possible to have a free group (in fact, a Finitely generated free group (?)) having a retract that is not a free factor. In other words, a retract of free group is not necessarily a free factor of free group.

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