Saving SAML response NameID as User attribute

Hello,

I connected keycloak with an Identity Provider that returns a username that can change in the future. The Identity Provider also provides an ID that never changes in the SAML Attributes, so I am using that attribute as the Principal Type to identify my users.

However I would like to save the username from the identity provider as a user attribute in keycloak. Is there a way to do this?