Carbocation Rearrangements in SN1 reactions
I think the solvolysis would involve rearrangement to the cyclopentyl carbocation as indicated in the figure below.  The driving force for the rearrangement is two-fold


without rearrangement, the cyclobutyl compound would generate a secondary carbocation; with rearrangement a more stable tertiary carbocation is generated instead
release of strain as the 4-membered ring opens to a 5-membered ring




The tertiary cyclopentyl carbocation ultimately captures a solvent molecule as indicated.