A week or eight days would be a "taste of Norway", especially if you enjoy outdoor adventure. More of the high mountain areas (Handagervidda) will be open in August than June.
I'm more of a winter visit to Norway and don't have first hand experience with the summer season, but June will be too early for the Hardangervidda. Our ski tour of the Hardangervidda started with our first night at the Tuva Hytte, a staffed full service hut. (Skied in from the Ustaoset train station). The Tuva's summer season from 1 July 2016 to 4 September 2016. Tuva is at 1200 meters ASL.
You can reach the Hardangervidda on the Oslo to Finse train line common to the Nutshell tour:
FROM HUT TO HUT ON THE PLATEAU In the summer, network with cairns and
red Ts is large and varied Hardangervidda. Also standard on the
cottages are varied. It features everything from simple stone arches
to large catering lodges with three course dinner. You can find them
on ut.no/hytte .
The Norwegian Trekking Association first cottage was Krokan at Rjukan
waterfall. Crofter was acquired in association's year of foundation in
1868 and opened as a tourist cabin in 1871. Tourist Association first
cabin up on the plateau was Krækkja , which opened in 1878.
Today there are about 50 lodges on Hardangervidda. Both service cabins
and privately owned lodges. Both contained with full board and
self-catering with food storage.
TOP HIKING Kinsarvik to Stavali (up the great Husedalen). Hjølmo to
Vivelid , Hedlo and Hadlaskard (in Veigdalen). Trondsbu / Tinnhølen
to Rauhelleren or Sandhaug . Halne - boat over Halnefjorden -
Rauhelleren . Tuva to Heinseter . Boat Mårbu from Synken within
Kalhovd . Haukeliseter to Hellevassbu and Litlos . Finse to Blåisen
. Stavsro to Gaustatoppen .
The most popular bike tour is from Haugastøl Flam. Bicycles can be
rented at Finse.
GOOD STARTING POINTS Haukeliseter , Kinsarvik , Hjølmo , Finse ,
Trondsbu / Tinnhølen , Halne , Haugastøl , Ustaoset, Tuva , Geilo,
Hakkesetstølen Solheimstulen , Mogen , Kalhovd and Rjukan .
Looking at the Tuva as an example of DNT huts:
The cabin is nicely situated entirely northeast of Hardangervidda. The
lodge is owned and operated by Aileen Flata Aker. The lodge provides
discount for DNT members. In winter run lodge as dining place from
mid-February until it begins to receive accommodation in March to
April. In the summer cottage welcomed overnight from late June [opens
01 July for 2016] to mid-September.