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It says quote "Unlike most familiars, her familiar can transform itself into a tattoo that she carries in her flesh. Transforming into a tattoo or back to normal familiar form is a move action for her familiar. In tattoo form, the familiar looks like a stylized version of itself, but does not count as a creature separate from the tattooed sorcerer. In tattoo form it continues to grant its special familiar ability, but otherwise has no abilities and can take no actions except to transform from tattoo into creature. A familiar tattoo cannot be erased or dispelled."
Note it saying special familiar ability. Does this mean that if the tattooed sorcerer were to have their tattoo familiar get the protector archetype, its protection effect would still be active even in tattoo form?
the "Special familiar ability" speak of the ability that the familiar grant to his master depending on it's type such as +4 to initiative or +2 to perception etc
which is marked at this table under "Special Ability".
the protector on the other hand replace some abilities the familiar get with feats and let him use them while in the master's space but they still need some kind of action to use, and a tattoo familiar in tattoo form get no action other then morphing back. also non of these other abilities are marked as special.
so beside the familiar's special ability from it's kind the protector familiar can't use it's archtype abilities while in tattoo form (unless noted otherwise).