We introduce a 1-complex $\mathrm{Wh}(\mathrm{\Gamma })$ associated with a finite regular cover $\mathrm{\Gamma }$ of the rose which is connected if and only if the fundamental group of the associated cover is generated by elements in a proper free factor of the free group. When the associated cover represents a characteristic subgroup of the free group, the complex admits an action of $\mathrm{Out}(F_n)$ by isometries. We then explore the coarse geometry of $\mathrm{Wh}(\mathrm{\Gamma })$. Every component of $\mathrm{Wh}(\mathrm{\Gamma })$ has infinite diameter, and the complex $\mathrm{Wh}({\mathbf{R}}_n)$ associated with the rose is nonhyperbolic. As corollaries, we obtain that the Cayley graph of the free group with the infinite generating set consisting of all primitive elements has infinite diameter and is nonhyperbolic.