You Do Need power plug adapter while traveling from Saint Barthélemy to Saint Helena.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Helena is G and in Saint Barthélemy are C and E. This is how they look like.

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E

Saint Helena

In Saint Helena, outlets of type G are in use.

Type G