They didn't sell it but rather split it. Every Motorola shareholder received 1 share of Motorola Mobility for each 8 shares of Motorola. After spinning off Motorola Mobility, they did a reverse 1-for-7 reverse stock split for Motorola and renamed it to Motorola Solutions.
It was initially the same owners for both companies but immediately diverged due to both being publicly traded. Google acquiring Motorola Mobility ended the historical shared ownership of the companies.