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In this paper we will investigate a family of five-stages high algebraic order embedded explicit six-step finite difference pairs, for the numerical solution of second order initial and/or boundary-value problems with periodical and/or oscillating solutions. Requiring the elimination of the phase-lag of the methods of the family of five-stages embedded explicit six-step algorithms and requiring also the elimination of the phase-lag's derivatives of the methods of the family of five-stages embedded explicit six-step schemes, we obtain a system of of equations. The solution of the previously obtained system of equations, leads to the determination of the free parameters of the new proposed finite difference pairs. Following the building of the new proposed finite difference pairs, we analyze theoretically the resulting schemes as follows: • We will study the local truncation error (LTE) of the resulting schems. • We will study the asymptotic form of the LTE which are produced using as model problem the radial Schrödinger equation. • We will study the comparison of the asymptotic forms of LTEs for several finite difference schemes of the same family. The comparison leads to the conclusions on the effectiveness of each member of the family of five-stages high algebraic order embedded explicit six-step finite difference pairs. • We will study the stability and the interval of periodicity of the resulting algorithms of the family of five-stages high algebraic order embedded explicit six-step finite difference pairs. • We will study the applications of the new builded family of five-stages high algebraic order embedded explicit six-step finite difference pairs to the numerical solution of several second order problems like the radial Schrödinger equation, astronomical problems etc. The previously described applications lead to the conclusions on the effectiveness of the members of the new proposed family of five-stages high algebraic order embedded explicit six-step finite difference pairs. [ABSTRACT FROM AUTHOR]