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The issue is not whether they have permissible purpose, or that the inquiry was unauthorized.
The issue is simply one of coding.
A current creditor has permissible purpose to obtain the credit report of their customers.
A party who requests a credit report must provide a statement of their permissible puprpose to the CRA.
The inquiry is then coded based on the stated permissible purpose.
Internal account reviews are, per CRA policy, provided a code that blocks the inquiry from being shown in credit reports they provide to others (i.e., it is coded as "soft").
The issue is simply improper coding of the inquiry if it appears in credit reports provided to others.
I would begin by contacting the CRA and requesting review of the coding. Either the creditor provided an improper code or statement of permissible purpose, or else the CRA improperly entered the appropriate code. I would not pursue as an unauthorized or improper inquiry, but as an incorrect administrative coding.