Last observation of the previous group

I would like to know, if I have data that I can group by a variable, how can I get the last observation of the previous group?

I have the following data:

dt <- data.table(a=c(1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5), b=sample.int(21))

I would like to create a new data.table that has the group ID and the difference between the last observation of the group from the last observation of the previous group. So that from the above I'd get:

   a  c
1: 1 NA
2: 2  9
3: 3  1
4: 4 -8
5: 5  5

Thanks!