Yesterday Cristiano Ronaldo was sent off in win over Barcelona:

• He was given a first yellow card for removing his shirt during the celebration of a goal.
• A couple of minutes later, he got the second yellow card for diving to attempt to win a penalty.

The second yellow card made him be sent off. After receiving the red card he pushed the referee. Such action normally involves a direct red card, but he had already received it.

So my question is:

After receiving a second yellow card, can a player get an "extra" red card for a blatant bad behaviour?

My impression is that a player in that situation is already out of the game, so any actions involving suspension will just be written down by the referee in the "certificate" of the game, so the federation will study it after the game and, eventually, penalise the player. However, I don't know if this is what rules say.

On 13th August 2017, Cristiano Ronaldo was sent off in win over Barcelona:

• He was given a first yellow card for removing his shirt during the celebration of a goal.
• A couple of minutes later, he got the second yellow card for diving to attempt to win a penalty.

The second yellow card made him be sent off. After receiving the red card he pushed the referee. Such action normally involves a direct red card, but he had already received it.

So my question is:

After receiving a second yellow card, can a player get an "extra" red card for a blatant bad behaviour?

My impression is that a player in that situation is already out of the game, so any actions involving suspension will just be written down by the referee in the "certificate" of the game, so the federation will study it after the game and, eventually, penalise the player. However, I don't know if this is what rules say.