I got an answer from the Ministry!!
The history is detailed in the following article: Krýsuvík kemst í eign Hafnarfjarðar published on 25 October 2019, which translates to Krýsuvík becomes the property of Hafnarfjörður.
Thus the area of Krýsuvík is incorporated in the Capital Region as Hafnarfjörður is one of its municipalities. Using Google Translate, this is the explanation:
Around 1930 it was difficult to farm around Hafnarfjörður; the townspeople were not self-sufficient in drinking milk and there was a shortage of pasture for sheep. No more arable land was obtained from Garðakirkjuland and other lands near the town were then being looked for. Krýsuvík was considered the most respectable and the town council of Hafnarfjörður wrote a letter to Einar Benediktsson in January 1933 and asked if the land in Krýsuvík or Herdísarvík was available for purchase. Two years later, the town had the option of buying Krýsuvík for ISK 50,000 and leased the land on voyages in 1935 for one year, with the intention of buying it later with data and quality.
By Act no. 11, 1 February 1936, the Treasury was authorized to expropriate lands in the jurisdiction of Hafnarfjörður and lands adjacent to Hafnarfjörður. After that, a great deal of controversy arose over the town's forthcoming purchase of the lands of Krýsuvík and Stóra-Nýjabær and the right of use. This went on for several years until the law was changed and law no. 101, effective 14 May 1940. It stated that Gullbringusýsla should receive its share of uncultivable pasture land in Krýsuvík and Stóra-Nýjabær for summer work for sheep according to an exchange agreement from 1 May 1939, but Hafnarfjörður would otherwise receive the lands with all the documents and benefits that came with them and should accompany them, with the exception of mining rights. On February 20, 1941, the Ministry of Justice and Ecclesiastical Affairs issued a waiver for Krýsuvík to Hafnarfjörður.