The Tottenham forward missed a late penalty – his second of the game – which would have taken the match to extra-time.
Kane levelled the scoring seven minutes into the second half when he dispatched a spot-kick past team-mate Hugo Lloris.
However, Olivier Giroud put the World Champions back in the lead with a header that deflected in off Harry Maguire.
Kane then had a late chance to score his second off the game when a VAR review showed Mason Mount was shoved to the ground.
But Kane ballooned his effort over the bar, much to the jubilation of the French crowd behind the goal.
And the forward says there’s ‘no hiding’ the fact that he should have scored from the spot, insisting he must take responsibility for the miss.
‘Absolutely gutted,’ said Kane on Instagram.
‘We’ve given it everything and it’s come down to a small detail which I take responsibility for.
‘There’s no hiding from it, it hurts and it’ll take some time to get over it but that’s part of sport.
‘Now it’s about using the experience to be mentally and physically stronger for the next challenge. Thanks for all the support throughout the tournament – it means a lot.’
Gareth Southgate will take time to consider his future following England’s exit.
His contract with England runs until 2024, meaning he would lead the Three Lions into the next European Championships.
However, Southgate insisted last night he will consider his options.
‘I think whenever I’ve finished these tournaments I’ve needed time to make correct decisions because emotionally you go through so many different feelings and the energy that it takes through these tournaments is enormous,’ Southgate said.
‘I want to make the right decision, whenever that is, for the team, for England, for the FA, and I’ve got to be sure whatever the decision I take is the right one.
‘I think it’s right to take a bit of time to do that because I know in the past how my feelings have fluctuated in the immediate aftermath of tournaments.’
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