Tunisian authorities have rounded up and arrested a group of suspects linked to Seifeddine Rezgui, the 23-year-old accused of killing 38 unarmed tourists on a beach in the coastal city of Sousse.

Tunisian interior minister Najem Gharsalli confirmed on Monday 29 June that authorities made the arrest after reports emerged that the gunman might have received help from others to plan the attack. Rezgui, a student at Kairouan university, shot dead the tourists, including at least 30 Britons, in the Tunisian beach resort.

"We have started by arresting a first group, a significant number of people, from the network that was behind this terrorist criminal," Gharsalli said. He did not elaborate further.

The father of the gunman, who is believed to have links with the Islamic State (Isis), said he is ashamed and shocked at his son's bloody acts.

"My god, I am so shocked," Hakim Rezgui said, according to the Guardian. "I don't know who has contacted him, influenced him or who has put these ideas in his head. He has new friends who got him into this."

In the aftermath of the shooting, UK prime minister David Cameron warned that IS militants were planning "terrible" attacks against Britain.

"There are people in Iraq and Syria who are plotting to carry out terrible acts in Britain and elsewhere and as long as ISIL (Islamic State) exists in those two countries we are at threat," Cameron told BBC radio.

Home secretary Theresa May, who visited the site of the attack, said there was no evidence that British tourists were specifically targeted.

May spoke at a joint press conference with Tunisian, French and German ministers. She said what happened on Friday was a "despicable act of cruelty".

"How could a place of such beauty, of relaxation and happiness be turned into such a scene of brutality and destruction?" she asked.

The Tunisian interior minister said that 1,000 extra armed police are being deployed in Tunisia following the attack. Up to 692 locations will be covered and armed guards are being put on duty inside hotels.

Britons killed in Tunisia beach attacks

Carly Lovett, 24, from Gainsborough in Lincolnshire

Sue Davey and her partner Scott Chalkley, both in their 40s and from Tamworth

Adrian Evans, his father, 78-year-old Patrick Evans and nephewJoel Richards, 19, from Wednesbury, West Midlands

Claire Windass, 54, from Hull, was on the beach with her husband, who survived

Jim and Ann McQuire, from Cumbernauld, North Lanarkshire

Trudy Jones, 52, from Blackwood, in Caerphilly county

Stephen Mellor, 59, from Bodmin in Cornwall, whose death was confirmed by his injured wife Cheryl

Bruce Wilkinson, 72, from Goole, East Yorkshire

Lisa Burbidge, in her 60s, from Whickham, Gateshead