dt <- data.table(grp = c(1,1,1,2,2,2), x = 1:6)
dt[, x_prev := shift(x = x, type = "lag", n = 1), by = grp]
dt[, x_next := shift(x = x, type = "lead", n = 1), by = grp]

print(dt)
     grp     x x_prev x_next
   <num> <int>  <int>  <int>
1:     1     1     NA      2
2:     1     2      1      3
3:     1     3      2     NA
4:     2     4     NA      5
5:     2     5      4      6
6:     2     6      5     NA

dt <- data.table(grp = c(1,1,1,2,2,2), x = 1:6)
dt[, `:=`(
  x_prev = shift(x = x, type = "lag", n = 1),
  x_next = shift(x = x, type = "lead", n = 1)
), by = grp]

print(dt)
     grp     x            x_prev            x_next
   <num> <int>            <list>            <list>
1:     1     1 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA
2:     1     2 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA
3:     1     3 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA
4:     2     4 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA
5:     2     5 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA
6:     2     6 NA, 1, 2,NA, 4, 5  2, 3,NA, 5, 6,NA

> sessionInfo()
R version 4.2.0 (2022-04-22)
Platform: x86_64-apple-darwin21.3.0 (64-bit)
Running under: macOS Monterey 12.4

Matrix products: default
LAPACK: /usr/local/Cellar/r/4.2.0/lib/R/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] data.table_1.14.3

loaded via a namespace (and not attached):
[1] compiler_4.2.0  httr_1.4.3      R6_2.5.1        generics_0.1.2  tools_4.2.0     lubridate_1.8.0