The character table for S_5 is
The character table for A_5 is here.
I followed the notations of the book, Fulton's "Representation Theory : A first course".
Note that the conjugacy class (12345) in S_5 with size 24 splits into two conjugacy classes (12345) and (21345) in A_5 with size 12, respectively. Note also that the conjugacy classes (12), (1234) and (12)(345) are not available in A_5.
Let H=A_5 and G=S_5. Then, by using Frobenius's reciprocity
Res U = U, Res U' = U, Ind U = U U',
Res V = V, Res V' = V, Ind V = V V',
Res W = W, Res W' = W, Ind W = W W', and
Res = Y Z, Ind Y = Ind Z = .
The last line is obtained by using the lemma,
where is the order of the conjugacy class in G of g,
are representatives of s conjugacy classes of H,
be the orders of these classes.