The Swedish prime minister, Ulf Kristersson, has said that he will allow the far-right Sweden Democrats (SD) into government for the first time – and give its members key ministerial posts – if his coalition wins the next general election.

Despite becoming Sweden’s second biggest political party after the Social Democrats in the last election, SD currently only play a supporting role in the minority-run coalition.

But Kristersson, who leads the centre-right Moderates, said on Wednesday that if his four-party coalition won September’s election SD would hold “big political influence and important ministerial posts within immigration and integration”.

In a joint press conference with SD’s leader, Jimmie Åkesson, Kristersson, whose polling after four years in government is not looking favourable, said: “We have agreed to jump-start the next term and form a strong majority government if we get the voters’ trust.”

The announcement was a watershed moment for Swedish politics, which to date has largely treated SD, a party with neo-Nazi roots, as a pariah.

Nooshi Dadgostar, leader of the Left party, said the prospect of the Sweden Democrats in government with the Moderates was “disgusting” and urged political leaders to “think again”.

“Now there is a lot at stake and now we know that we can have right-wing extremist ministers in the government,” she told Dagens Nyheter. “Now we have to come together to offer a different path for Sweden.”

Since replacing the Moderates as Sweden’s second biggest party in 2022 and entering into a supporting role in Kristersson’s coalition, the SD’s policies have had considerable influence on the government – particularly on immigration.

Their rhetoric has shaped policy across the political spectrum, including on the left-leaning opposition, the Social Democrats, who have also adopted hardline immigration and integration policies, like their Danish counterparts under Mette Frederiksen.

Standing by Kristersson’s side on Wednesday, Åkesson, who has led the SD since 2005, when it was a small party on the fringes of Swedish politics, said that after the next election he expected his party to get influence proportional to its size. “We have been clear that after the next election we will either be a governing party or an opposition party,” he said.

SD’s policies include stopping people from countries outside Sweden’s “immediate area” from claiming asylum in the Scandinavian country, a step that would contravene human rights law, and ensuring that “more of those who do not have the right to be in Sweden to leave than come to Sweden”.

Mass immigration to Sweden, the party claims, has “changed Sweden for the worse” and resulted in “many societal problems”. Islamophobia is also prominent. In a recent documentary, Åkesson claimed that to be Muslim and to be Swedish was “a contradiction”.

Wednesday’s announcement comes after the leader of the ailing Liberals, Simona Mohamsson, shocked her party last month by abruptly switching positions on SD, announcing that her party would accept being in government with a far-right force she had previously denounced as racist. She also held a press conference alongside Åkesson in which they hugged.

Magdalena Andersson, the leader of the Social Democrats and former prime minister, said Kristersson and Åkesson’s proposal would lead to a “historically weak prime minister”.

“It is obvious that it is Jimmie Åkesson who is holding the baton,” she told broadcaster SVT. “In an organisation where the person who is formally the boss is not the real boss, there will be instability and an organisation without the power to act.”

The left-wing opposition parties have so far not presented a proposal for an alternative government.