google box and back and forth arrows have disappeared from header. How can I restore them?

Welcome to Apple Support Communities. We're users here and don't speak for "Apple Inc."

First check to see that Safari, View, Show Toolbar is checkmarked.

If it is, then try Safari, View, Customize Toolbar, then click and drag the box around ...or drag the default set onto the toolbar. or click-on and drag just the missing items to your toolbar.