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In this paper, we analyze the natural homomorphism from the genus ghandlebody group to the outer automorphism group of the free group with rank g,in terms of length spectra. In general, the preimage of each fully irreducibleouter automorphism contains a potentially infinite number of pseudo-Anosovmapping classes. Our study reveals a crucial relationship: for anypseudo-Anosov map in this preimage, its stretch factor must equal or exceedthat of the corresponding fully irreducible outer automorphism. Notably, in thecase of genus two, we establish that the minimum stretch factor among thesepseudo-Anosov maps is less than ten times the stretch factor of the fullyirreducible outer automorphism. These results partially address a question byHensel and have practical implications, including a lower bound for thegeodesic counting problem in the genus two handlebody group.

Uniform difference between the length spectra of Out(F2) and the genus two handlebody group