Any bus companies offer transport from Reykjalið (Mývatn) to Askja? Or maybe Hitchhiking?
The Rough Guide Iceland (David Leffman, James Proctor) gives these details.
Tours from Reykjalið
If you don't want to take the following tours, you'll need to hire a car or bicycle from Hótel Reynihlíð or make use of the eight-kilometre-long Reykjalið-Grjótagjá-Hverfell Dimmuborgir hiking trail. The following excursions run in summer only; either contact operators direct or book through Mývatn's Visitor Centre.
SBA buses (www.sba.is) runs return day trips from Akureyri round the lake and out to Krafla (9500kr), and an extended version out to Dettifoss in Jökulsárgljúfur National Park (10,500kr). It also operates three-day expeditions to ice caves and glacier traverses at Kverkfjöll and Vatnajökull (25,000kr; see Routes to Kverkfjöll).
The track south from Mývatn to the Interior at Askja is covered by bus with Mývatn Tours (464 1920, www.askjatours.is; 14,000kr); this is a lengthy day-trip, though you can arrange in advance to stay at huts along the way and be collected another day. Mývatn Adventures (464 4164, www.myvatn.is; 28,000kr) does the same thing in super-jeeps, and also heads right down to the fringes of the Vatnajökull ice cap at Lofthellier Ice Caves (June-Sept daily; 7700kr), frozen lava caverns discovered in 1989 (35,000kr).
For scenic flights, Mýflug Air (464 4400, www.myflug.is) out of Reykjalið can take you for a spin over Mývatn and Krafla, or out to Askja and Dettifoss, or out to Grimsey - trips last from twenty minutes to two hours and cost 10,000-39,500kr. Finally, you can arrange horseriding in the area through Hlið (464 4103).