Nicola Sturgeon has said anti-English nationalists are not welcome in the SNP.
In a robust defence of her vision of an inclusive pro-independence movement, the First Minister suggested her party had to be vigilant in rooting out extremists.
Several senior SNP figures - including Justice Secretary Humza Yousaf - have admitted they are uncomfortable with the international connotations of the world 'nationalist'.
READ MORE: Sturgeon at the fringe in full
Ms Sturgeon, speaking to comedian Matt Forde at the Fringe, insisted her politics were internationalist.
Asked about nationalism, she said: "I have some problems with that word because of the global connotations of it but first and foremost I'm an internationalist - I want Scotland to be independent to give us more opportunity to chart our own course domestically but also so that Scotland can co-operate and play a bigger role with other countries on the world stage."
Ms Sturgeon added: "The SNP today is the most pro-immigration party in the UK. My nationalism is rooted in a desire to make the country I live in as good as it can be.
"I don't care where you come from, if you want to live in Scotland and consider yourself Scottish that's fine by me."
Ms Sturgeon recently encouraged immigration from England - with cross-border migration contributing to Scotland's population growth. The Yes movement includes individual people from England and a group called English Scots for Independence.
READ MORE: Humza Yousaf says he struggles with the name of the SNP and its connotations
The First Minister spoke about so-called cybernats and said there was an "unrealistic expectation" for political leaders to "police" supporters' behaviour and comments on social media.
She said: "I'm not responsible for everything people say on Twitter, thank God.
"But it cuts both ways - I try not to look at it but some of the abuse I get on Twitter would literally make your hair curl. "It's horrible, misogynist, really filthy, disgusting stuff."
Asked about inadvertently inspiring people who are racist, Ms Sturgeon said: "Any party, any movement - regardless of what they stand for - will attract people you don't want."
Her remarks came after a controversial blogger, Stuart Campbell of Wings Over Scotland, attacked Ms Sturgeon's leadership and said he was mooting the creation of an alternative pro-independence party.
Mainstream Scottish nationalists have criticised Mr Campbell for both his tone - he is influential among "cybernats" - and his politics and warned against a shift to US-style populism.
Some unionists have sought to portray Mr Campbell's brand of aggressive nationalism - or various other fringe groups and figures - as symptoms of what they see as a malaise within wider movement.
READ MORE: Wings over Scotland blogger attacks SNP and Greens in Yes movement split
Mr Campbell said he wanted to push the SNP in to holding a second independence referendum in this Holyrood term. Speaking to the BBC earlier this month, he said:
He said: “I don't think it's a great secret that I, and a lot of other people, are rather uncomfortable with the direction that the SNP is moving in at the moment."
Many visitors to Edinburgh during the Festivals have seen a banner calling for 'England to get out of Scotland'. Ms Sturgeon sought to distance herself from such sentiments. She said: "The person with that banner does not speak for the SNP. That kind of sentiment has no place in Scotland."
She added: "You can be absolutely vehement and resolute about calling it out and saying very clearly to people: 'If that's your opinion, you don't belong in this party and we don't want you.'
"People who put up banners like that, I don't want them in the SNP.
"The most important thing for a person in my position is not to pretend that my party will never attract people with those kind of opinions but to be absolutely clear that we will never make the SNP a comfortable or welcoming place for them."
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