What component of the cell membrane connects signals from the outside of the cell with the inside of the cell and vice versa?

Integrins.

Explanation:

Integrins may be defined as the important protein that helps in the attachment of cell cytoskeleton to the extracellular matrix. These proteins play an important role in the cell communication.

The signals and the communication occurs between the cell and the extracellular matrix. They work as the adhesion receptor for the different extracellular ligands and transfer the biochemical signal in the cell by using the downstream effector protein. Hence, they can transmit the information between inside and outside of the cell.

Thus, the answer is integrin.