I'm assuming that the Hero Association thought it was King and not Saitama who killed Elder Centipede, otherwise they would have likely promoted him. But how about Bang, Bomb and Genos who were there? Do they know who killed Elder Centipede or couldn't they tell as Elder Centipede was charging against King and in a flash Saitama killed him?
The credit for defeating Elder Centipede Centichoro probably went to king like how he usually gets it for the monsters defeated by Saitama. But in the matter of Genos, Bang and Bomb knowing who actually defeated Centichoro, Genos knew it was Saitama.
Because as soon as the centipede disintegrated, Genos asked Saitama what is he lacking again?
"What is it that master (Saitama) has and I don't?".
And Saitama reply.
"Power I guess"
This means that Genos knew Saitama was the one and wanted to know how is he able to defeat these monsters so easily.
As for Bang and Bomb, they are quite exceptional martial artists. Their sense of everything is extraordinary. So I doubt that they didn't realize it was Saitama who finished off the Elder Centipede.
Moreover, Bang already has an idea of Saitama's strength because of the meteor incident and also, from the time Bang and Saitama were competing over some games at the dojo.
On the show Bang, Bomb, and Genos are only standing a short distance away and were all staring at Saitama who was still in a punch pose after destroying Elder C. They would definitely have known Saitama killed it-though no doubt King/Genos/Bang/Bomb would all get more credit for the actual kill with Saitama getting supporting credit.
I mean if you seen the current episode then it should be really clear that it showed Genos shocked again seeing Saitama in the place, so i guess that implicates that he saw him killing the centipede because he also asked saitama what he lacked and I think Bang must have also known that it could only he Saitama who could have destroyed the monster as he is well aware of his extraordinary powers