Anyone in possession of a gun in Britain must have the correct permit. There are essentially four variations:
The Firearm Certificate
this is issued to U.K. residents, covering rifles and/or airguns. This certificate will list the firearm(s) possessed, and those allowed to be purchased or acquired, together with the quantity of ammunition that may be held, purchased or acquired. There may also be strict limitations on exactly where the firearm(s) may be used. This certificate will not entitle the holder to purchase automatic, semi-automatic (other than .22 rimfire), or pump-action (other than .22 rimfire) rifles. Any air rifle with a muzzle energy greater than 12 ft/lbs, or air pistol greater than 6 ft/lbs, may not be purchased unless you possess a Firearms Certificate authorising you to do so. Handguns are now effectively banned.
The Shotgun Certificate
will show the names and serial numbers of any shotguns possessed. There are currently no restrictions on the acquisition of shotguns, providing the details of any transaction are noted on the certificate, and the issuing police force informed. It is necessary to produce your certificate when purchasing cartridges. For the purposes of the certificate, a shotgun is defined as a smoothbore gun, with barrel(s) at least 24" (610mm) long. Semi-automatics must have a magazine capacity of no more than two shots: this must be a permanent restriction, verified by either the London or Birmingham Proof Houses.
The Visitors Firearm Permit
will show the make and serial number of any firearm that a visitor to Britain is permitted to bring in to the country with them, or borrow whilst here. It is not possible to purchase a firearm with a Visitors Permit, but ammunition may be acquired if such authorisation is shown. Limitations on the type of firearm that you import are the same as for U.K. residents; and restrictions will apply as to where the firearm may be used.
The Visitors Shotgun Permit
lists any shotguns that the visitor may bring into the country. These shotguns are defined as for the U.K. resident; therefore it is extremely unlikely that any semi-automatic will qualify, unless it has already been to Britain, and been approved by the Proof House. Any visitor wishing to purchase or acquire a shotgun whilst in Britain, must ensure that the police force issuing the permit put the necessary "authority to purchase" on that permit, which must also be produced when purchasing cartridges.
FAQs on Firearms licensing can be found here: http://www.met.police.uk/firearms_licensing/faqs.html