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
I think they all heard King scream for Saitama. Then, when the Elder Centipede is killed, Saitama is standing in a post punch stance and King is cowering on the ground. I doubt Saitama will take full credit. He's known all season King is getting credit for his kills and doesn't do anything to correct it. And, since King is the only one to beat him, even if only in video games, he's not going to blow King's cover. I think he considers King a friend and someone he respects. Genos is going to do as Saitama says. Bang already knew and likely told Bomb. They will respect Saitama's wishes. I think the Hero Association will promote Saitama because he was the only non-S-Class hero left standing, but he won't be moved to the #1 spot.
I think Bang and Genos know that Saitama defeated the monster, because they know the power of Saitama. But as for Bomb, I don't know… Moreover, I don't think King would even take the credit for defeating Elder Centipede, as it was shown that he felt shame for taking credit from Saitama for defeating several monsters.