Estimating the percentage of common set elements over combined past periods in a panel

Question

I have a time-series panel dataset that is structured in the following way: There are 2 funds that each own different stocks at each time period.

df <- data.frame(
  fund_id = c(1,1,1,1,1,1,1,1, 1, 2,2,2,2),
  time_Q = c(1,1,1,2,2,2,2,3, 3, 1,1,2,2),
  stock_id = c("A", "B", "C", "A", "C", "D", "E", "D", "E", "A", "B", "B", "C")

)

> df
   fund_id time_Q stock_id
1        1      1        A
2        1      1        B
3        1      1        C
4        1      2        A
5        1      2        C
6        1      2        D
7        1      2        E
8        1      3        D
9        1      3        E
10       2      1        A
11       2      1        B
12       2      2        B
13       2      2        C

For each fund, I would like to calculate the percentage of stocks held in that current time_Q that were also ever held in any of the previous one to 2 quarters. So basically for every fund and every time_Q, I would like to have 2 columns with past 1 time_Q, past 1-2 time_Q which show what percentage of stocks held in that time were also present in any of that past time_Qs. Here is what the result should look like:

result <- data.frame(
  fund_id = c(1,1,1,2,2),
  time_Q = c(1,2,3,1,2),
  past_1Q = c("NA",0.5,1,"NA",0.5),
  past_2Q = c("NA",0.5,1,"NA",0.5)
)

>  result
  fund_id time_Q past_1Q past_1_2Q
1       1      1      NA      NA
2       1      2     0.5     0.5
3       1      3       1       1
4       2      1      NA      NA
5       2      2     0.5     0.5

I already asked a similar question here, but now I'm looking for common elements across any of the past lagged periods. I'm looking for a dplyr or data.table scalable solution where I can have around 12 past quarters and deal with multiple funds and stocks and time periods.

Thanks in advance!