The hardline faction, which includes Brexit Secretary David Davis, Defense Secretary Michael Fallon and Environment Secretary Michael Gove, wants a clean break with the European Union in order to regain full control over Britain's borders and do trade deals with non-EU countries.
In this week's Headliner, elder statesman and leading eurosceptic Norman Lamont defends the Conservative Party and its plans for Brexit.
The Prime Minister's uncompromising approach to the negotiations might not be the one her Government now follows. "I think it will not actually make as dramatic a difference as people think". The party has refused to give a time frame for reaching a deal, though May is due in Brussels for an European Union summit on June 22-23 when she will want to show she has a solid grip on power.
In her letter triggering the two-year Brexit process on March 29, May had insisted that Britain wanted to discuss them in parallel.
"To change the leader of the party now would simply be to pile on the chaos", says Norman Lamont, an ardent Brexit supporter and former chancellor of the exchequer, who believes May's re-election, despite the loss of her majority, affirms her mandate. Mr Hammond also warned about failing to get a Brexit deal.
Adding to what Queen Elizabeth II called the "sombre national mood" have been three terrorist attacks in three months and a fire in a London tower block in which 58 people are presumed dead.
Having lost a majority in parliament, May is in talks with Northern Ireland's Democratic Unionist Party to secure the support of its 10 lawmakers to win any kind of vote, including on the pieces of legislation needed to enact Britain's divorce from the EU.
- June 22 - Mr Barnier reports back to the leaders of the EU27, who are expected to endorse a process for relocation of European Union agencies now based in the UK.
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Other members of May's Conservatives have called for a more inclusive approach on Brexit strategy that would include the voices of opposition parties as well as the views of Scotland and Northern Ireland, which both voted to stay in the EU.
Gabriel said "it would naturally be best if Britain didn't leave at all". "That is why we are pushing ahead with negotiations on Monday".
The Government agreed with the need to focus on the rights of European Union citizens living in the United Kingdom and British nationals living on the continent.
Addressing her party on Monday after one of its most memorable electoral failures, May said she would take a broader, more consultative approach to the Brexit talks.
An EU relationship similar to Norway's is the only model that could attract public and political support in Britain, without threatening EU principles or inflicting serious economic costs on either side.
European Union officials played down the importance of Britain's lack of a clear final plan, saying that talks on other issues can go ahead without deciding yet on a new relationship.
Britain's finance minister said Friday that jobs and growth should be the priority when Brexit negotiations begin next week, indicating that the weakened government in London might be softening its tone with Brussels.