Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme?

I just know that the category of etale group schemes over a k is equivalent to the category of abstract groups with a galois action. So I am wondering what the objects would be like in the left category. In fact any finite etale scheme over k should be a finite union of spectrums of fields which are finite separable extensions of k, and if it is a group scheme then it should have a rational point, but I don't see how one can translate the rational point to other points, so I guess there might be some non-rational point in the group scheme. is that right? an example?