Here is what Wikipedia says about these islands:
The Kerguelen Islands, also known as the Desolation Islands, are a group of islands in the southern Indian Ocean constituting one of the two emerged parts of the mostly submerged Kerguelen Plateau. The islands, along with Adélie Land, the Crozet Islands and the Amsterdam and Saint Paul Islands are part of the French Southern and Antarctic Lands and are administered as a separate district. There are no indigenous inhabitants, but France maintains a permanent presence of 50 to 100 scientists, engineers and researchers.
I could not see any flag on the Ile de Brehat, but this is how the Kerguelen Islands flag looks:
From another source I found out that the French Government in 1986 started a registry at Kerguelen Islands. (The ships could be operated with a crew consisting of only 25% French nationals.)
The Ile de Brehat is owned by ALDA Marine Services S.A.S, operated by Luois - Dreyfus Armateur and apparently working for Alcatel-Lucent.
The ship is now on its way from Baltiysk (in the Kaliningrad area) to Everingen. (It would be interesting to know what type of cables the Ile de Brehat has been laying in Baltiysk, a major naval base of Russia´s Baltic Fleet.)
Ship facts:
Built: 2002 by Hyundai Mipo Dockyard, Ulsan, South Korea
Length: 140.36 m
Breadth: 23.4m
Accomodation for 70 crew members