Indeed - but we are all agreed that what the OP has done is unacceptable - there must be a 13A fuse somewhere between the feeding circuit and the 4 x 13A 'outlets'.What size is the cable in the extension lead ? What is it's maximum safe load current ? As it was designed to be used with a plug fitted with a fuse 13 amp or less it may be that 13 amps is the maximum. If it is a cheap import it might be a lot less than 13 amps. ... It is ( as mentioned before ) possible that the load could be as much as 4 x 13 amps. Even a good quality extension lead cable will heat up with that load.
As eric has pointed out, it might have to be an RCD FCU, since the extension is effectively become a number of hard-wired sockets.Fit an FCU with a 13 amp fuse next to the socket and feed the extension from that. Use a dual box that will take the socket and the FCU on the one back box.
Kind Regards, John