You Do Need power plug adapter while traveling from Nauru to Faroe Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Faroe Islands are C, E, F and K and in Nauru is I. This is how they look like.

Nauru

In Nauru, outlets of type I are in use.

Type I

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K