There's no (single) 'official' day for the start of any season. Different people use different definitions.
For example (since the sun reaches its highest point in the sky around that date), midsummer's day (aka the summer solstice) is on June 21st, with the winter solstice (with the sun at its lowest point in the sky) being on December 21st. Assuming all of the seasons to be of equal length, each season must about 91 days long, so the start and finish of summer must therefore be roughly 45 days either side of June 21st, making the last day of summer around August 12th.
However while that method of deciding when each season starts and ends might find favour with astronomers, it doesn't fit very well with most people's perceptions of when the seasons actually occur. Meteorologists have a far simpler way of looking at it: June, July and August are the warmest months in the UK, so those three months fall into 'summer'. That means that September, October and November form 'autumn'. December, January and February make up 'winter', while March, April and May are grouped together as 'spring'.