How to perform Mulliken population analysis by hand with GAMESS output?
Using ORCA and reverse engineering the calculation provided in the question (HF/STO-3G at a $\ce{H}$-$\ce{F}$ distance of 0.955464 Angstroem), I obtained the following output, to which I added the matrix product of the density and overlap matrices, dubbed the Mulliken Population Matrix. By comparing to the output of other programs, the reader can follow the analysis.

------------------
MOLECULAR ORBITALS
------------------
                  0         1         2         3         4         5
             -25.90350  -1.45985  -0.57365  -0.46312  -0.46312   0.58982
               2.00000   2.00000   2.00000   2.00000   2.00000   0.00000
              --------  --------  --------  --------  --------  --------
0H   1s        -0.005340 -0.150428  0.533708 -0.000000  0.000000 -1.054338
1F   1s         0.994746  0.250679  0.078267 -0.000000  0.000000 -0.080569
1F   2s         0.022260 -0.946706 -0.410898  0.000000 -0.000000  0.515847
1F   1pz       -0.002673  0.078253 -0.698059  0.000000 -0.000000 -0.816434
1F   1px        0.000000 -0.000000 -0.000000 -0.998068 -0.062124 -0.000000
1F   1py       -0.000000  0.000000 -0.000000 -0.062124  0.998068  0.000000


-------
DENSITY
-------
              0          1          2          3          4          5
  0       0.615003  -0.002499  -0.154014  -0.768633  -0.000000   0.000000
  1      -0.002499   2.116969  -0.494672  -0.075355   0.000000  -0.000000
  2      -0.154014  -0.494672   2.131170   0.425379   0.000000   0.000000
  3      -0.768633  -0.075355   0.425379   0.986834  -0.000000   0.000000
  4      -0.000000   0.000000   0.000000  -0.000000   2.000000   0.000000
  5       0.000000  -0.000000   0.000000   0.000000   0.000000   2.000000

--------------
OVERLAP MATRIX
--------------
              0          1          2          3          4          5
  0       1.000000   0.045145   0.423474  -0.335755   0.000000   0.000000
  1       0.045145   1.000000   0.237990  -0.000000   0.000000   0.000000
  2       0.423474   0.237990   1.000000   0.000000   0.000000   0.000000
  3      -0.335755  -0.000000   0.000000   1.000000   0.000000   0.000000
  4       0.000000   0.000000   0.000000   0.000000   1.000000   0.000000
  5       0.000000   0.000000   0.000000   0.000000   0.000000   1.000000

--------------------------
MULLIKEN POPULATION MATRIX
--------------------------

 0,807741631  -0,0113884814   0,1058290434   -0,9751233323  0  0
-0,091108347   1,9991291934   0,0080871908   -0,0745159483  0  0
 0,583325991   0,0055721863   1,9482220861    0,4770899706  0  0
-0,923232404  -0,0088189886   0,0819491725    1,2449063729  0  0
 0             0              0               0             2  0
 0             0              0               0             0  2

-----------------------
MULLIKEN ATOMIC CHARGES
-----------------------
0 H :    0.192259
1 F :   -0.192259
Sum of atomic charges:    0.0000000