The search of the Second Foundation (SF) was organized first by the Mule himself. He successfully defeated the Fist foundation. We know that. He dreamed of enlarging his dictatorships to both the Foundations to make his victory complete and to rule the entirely Galaxy. He knew that the Second Foundationers, too, have a unique mental ability to tap minds. The Mule has got the same ability. So through the first part of the book you’ll notice the battle of mind tapping.
Such that to find where the SF is situated, the Mule sends his two men Pritcher and Channis. Pritcher is converted or mentally manipulated by the Mule. Channis is not. He is a devoted dog. That was where the Mule lost. Channis was of the SF men. The mule tried to kill him when he learnt Channis was a spy but Channis knew how to deter the Mule.
He warned the Mule that the instant before he was killed he would remove the Mule’s manipulation from Pritcher who was, before it, a lieutenant fighting the Mule. It means Pritcher free from manipulation would kill the Mule in return. So it’s better not to kill Channis in the first place. It saved Channis. In this moment the main psychologist of the SF, The First Speaker intervened. He made certain changes in the Mule’s mind so that he became not an aggressor, but a benevolent person.the five years of the Mule’s epoch ended. After a long period the ruined first Foundation was back to life somehow.
The second search was done by the first Foundation. the first Foundationers grew envious over the SF. They considered the latter as enemies. While the SF showed no hostility towards the first Foundation. They even helped them in defeating The First Citizen of Kalgan who sent his armada to ruin Terminus the capital planet of the first Foundation.
The SF lead Hari Seldon’s plan to create Encyclopedia and the Second Empire out of ruins of the first one. It was not interested to kill the same people who were the part of the plan. All the same the first Foundation didn’t like the SF. They searched and thought they’ve located its place. They were wrong but they didn’t know about it and they will not know forever. They will be thinking they isolated the SF in the planet Kalgan. But they are not there. They are where the first Galactic Empire had been once prospered. On the Trantor, the birthplace of Hari Seldon.